text: a high-pressure system will decay. It is by getting the underlying physics right that we are able to achieve some level of positive skill in any forecast attempt (Fig. 1). The macroeconomic components of IAMs do not offer true forecasts that can be assigned a skill score. Rather they reﬂect expert opinions (Moss et al., 2010) and are mostly unconstrained by external physical forces since they are policy-driven and mathematically constructed so as to allow for an extremely broad range of possible futures (Pindyck, 2013). They consider labor, physical capital, and human inspired technological change to be the motive forces for economic production and growth. The focus is on individuals, nations, and economic sectors. The model equations describe how physical capital and human prosperity grow with time, and how energy choices tie in with greenhouse gas emissions (Solow, 1956; Nordhaus and Sztorc, 2013). Part 1 of this study (Garrett, 2014, hereafter referred to as Part 1) described a second, more deterministic approach. CO is considered to be long-lived and well mixed in the atmosphere, so the magnitude of greenhouse forcing is almost 2 entirely unrelated to the national origin of anthropogenic emissions. Then, civilization can be described as a whole, one where small-scale details at personal, regional, or sectoral levels are not treated explicitly. The only quantity in the model that needs to be resolved is an aggregated global economy that is inclusive of all civilization elements, including human and physical capital combined. As an alternative to IAMs, this new approach offers a means for integrating human systems and physical systems under a common framework, one where the governing equations are consistently derived from ﬁrst thermodynamic principles. Much like the primitive equations of a prognostic weather model, global economic growth is expressed as a non-equilibrium response to external gradients driving energy dissipation and material ﬂows. There is no explicit role for human decisions; physics does not readily allow for mathematical expressions of policy. Rather, economic innovation and growth is treated primarily as a geophysical phenomenon, in other words the totality of civilization is expressed as an emergent response to available reserves of raw materials and energy supplies. Whatever the approach that is applied, it is important that any societal model be evaluated for performance. Weather, climate, and ﬁnancial models are regularly evaluated through hindcasts or backtesting. Economic models that simulate the long-run development of humanity need not be an exception. A good model, even one that includes policy, should be able to reproduce current events with positive skill starting at a point some decades in the past. To beat the zero-skill hindcast of persistence, the model would invoke ﬁts to concurrent trends to the minimum extent possible. In the period following World War II, an economic “front” passed that propelled civilization towards unprecedented levels of prosperity and, by proxy, greenhouse gas emissions. This paper examines whether the theoretical model introduced in Part 1 can explain the evolution of this front. Section 2 outlines the philosophical and thermodynamic basis for describing economic evolution with physics. Section 3 evaluates this model from hindcasts. Sections 4 and 5 discuss and summarize the results. 2 Forces for economic growth 2.1 The relationship of energy dissipation to human wealth A formal framework for the non-equilibrium thermodynamics of civilization growth was laid out in Part 1 (Garrett, 2014). The basis for a model linking economics to physics is a fundamental identity that relates a monetary expression for wealth to the rate at which civilization powers itself with primary energy sources (Garrett, 2011). We all have some sense of the difference between civilization and its uncivilized surroundings since we have farms, buildings, human population, vehicles, and communication networks. As shown in Fig. 2 and discussed in Appendix A, this difference implies the existence of a gradient between civilization and its environment. Gradients allow for irreversible thermodynamic ﬂows. A consumption and dissipation of potential energy by civilization sustains internal reversible circulations within civilization that characterize all its activities. The hypothesis that was made in Part 1 is that civilization is effectively a heat engine whose power can be rep- resented in more human terms as economic wealth. Absent any energy consumption, civilization would necessarily decay towards an uncivilized, worthless equilibrium where the gradient ceased to exist and all internal circulations stopped. Wealth is able to grow only when net work is done to grow into the environment. Real economic production occurs when raw materials can be incorporated into civilization’s structure faster than civilization decays (Fig. 2). Growth at a net positive rate expands civilization’s interface with reserves of energy. It enlarges civilization, creating new wealth and a greater overall capacity of civilization to consume and sustain internal circulations. Expressed as an integral, current economic wealth is the net accumulation of past net physical work and real economic production. In effect, there is no intrinsic wealth in and of itself. Rather, as discussed in Part 1, wealth is built from a network 2

text: a

text: high-pressure

text: system

text: will

text: decay.

text: It

text: is

text: by

text: getting

text: the

text: underlying

text: physics

text: right

text: that

text: we

text: are

text: able

text: to

text: achieve

text: some

text: level

text: of

text: positive

text: skill

text: in

text: any

text: forecast

text: attempt

text: (Fig.

text: The

text: macroeconomic

text: components

text: of

text: IAMs

text: do

text: not

text: offer

text: true

text: forecasts

text: that

text: can

text: be

text: assigned

text: a

text: skill

text: score.

text: Rather

text: they

text: reﬂect

text: expert

text: opinions

text: (Moss

text: et

text: al.,

text: and

text: are

text: mostly

text: unconstrained

text: by

text: external

text: physical

text: forces

text: since

text: they

text: are

text: policy-driven

text: and

text: mathematically

text: constructed

text: so

text: as

text: to

text: allow

text: for

text: an

text: extremely

text: broad

text: range

text: of

text: possible

text: futures

text: (Pindyck,

text: They

text: consider

text: labor,

text: physical

text: capital,

text: and

text: human

text: inspired

text: technological

text: change

text: to

text: be

text: the

text: motive

text: forces

text: for

text: economic

text: production

text: and

text: growth.

text: The

text: focus

text: is

text: on

text: individuals,

text: nations,

text: and

text: economic

text: sectors.

text: The

text: model

text: equations

text: describe

text: how

text: physical

text: capital

text: and

text: human

text: prosperity

text: grow

text: with

text: time,

text: and

text: how

text: energy

text: choices

text: tie

text: in

text: with

text: greenhouse

text: gas

text: emissions

text: (Solow,

text: Nordhaus

text: and

text: Sztorc,

text: Part

text: of

text: this

text: study

text: (Garrett,

text: hereafter

text: referred

text: to

text: as

text: Part

text: described

text: a

text: second,

text: more

text: deterministic

text: approach.

text: CO

text: is

text: considered

text: to

text: be

text: long-lived

text: and

text: well

text: mixed

text: in

text: the

text: atmosphere,

text: so

text: the

text: magnitude

text: of

text: greenhouse

text: forcing

text: is

text: almost

text: entirely

text: unrelated

text: to

text: the

text: national

text: origin

text: of

text: anthropogenic

text: emissions.

text: Then,

text: civilization

text: can

text: be

text: described

text: as

text: a

text: whole,

text: one

text: where

text: small-scale

text: details

text: at

text: personal,

text: regional,

text: or

text: sectoral

text: levels

text: are

text: not

text: treated

text: explicitly.

text: The

text: only

text: quantity

text: in

text: the

text: model

text: that

text: needs

text: to

text: be

text: resolved

text: is

text: an

text: aggregated

text: global

text: economy

text: that

text: is

text: inclusive

text: of

text: all

text: civilization

text: elements,

text: including

text: human

text: and

text: physical

text: capital

text: combined.

text: As

text: an

text: alternative

text: to

text: IAMs,

text: this

text: new

text: approach

text: offers

text: a

text: means

text: for

text: integrating

text: human

text: systems

text: and

text: physical

text: systems

text: under

text: a

text: common

text: framework,

text: one

text: where

text: the

text: governing

text: equations

text: are

text: consistently

text: derived

text: from

text: ﬁrst

text: thermodynamic

text: principles.

text: Much

text: like

text: the

text: primitive

text: equations

text: of

text: a

text: prognostic

text: weather

text: model,

text: global

text: economic

text: growth

text: is

text: expressed

text: as

text: a

text: non-equilibrium

text: response

text: to

text: external

text: gradients

text: driving

text: energy

text: dissipation

text: and

text: material

text: ﬂows.

text: There

text: is

text: no

text: explicit

text: role

text: for

text: human

text: decisions;

text: physics

text: does

text: not

text: readily

text: allow

text: for

text: mathematical

text: expressions

text: of

text: policy.

text: Rather,

text: economic

text: innovation

text: and

text: growth

text: is

text: treated

text: primarily

text: as

text: a

text: geophysical

text: phenomenon,

text: in

text: other

text: words

text: the

text: totality

text: of

text: civilization

text: is

text: expressed

text: as

text: an

text: emergent

text: response

text: to

text: available

text: reserves

text: of

text: raw

text: materials

text: and

text: energy

text: supplies.

text: Whatever

text: the

text: approach

text: that

text: is

text: applied,

text: it

text: is

text: important

text: that

text: any

text: societal

text: model

text: be

text: evaluated

text: for

text: performance.

text: Weather,

text: climate,

text: and

text: ﬁnancial

text: models

text: are

text: regularly

text: evaluated

text: through

text: hindcasts

text: or

text: backtesting.

text: Economic

text: models

text: that

text: simulate

text: the

text: long-run

text: development

text: of

text: humanity

text: need

text: not

text: be

text: an

text: exception.

text: A

text: good

text: model,

text: even

text: one

text: that

text: includes

text: policy,

text: should

text: be

text: able

text: to

text: reproduce

text: current

text: events

text: with

text: positive

text: skill

text: starting

text: at

text: a

text: point

text: some

text: decades

text: in

text: the

text: past.

text: To

text: beat

text: the

text: zero-skill

text: hindcast

text: of

text: persistence,

text: the

text: model

text: would

text: invoke

text: ﬁts

text: to

text: concurrent

text: trends

text: to

text: the

text: minimum

text: extent

text: possible.

text: In

text: the

text: period

text: following

text: World

text: War

text: II,

text: an

text: economic

text: “front”

text: passed

text: that

text: propelled

text: civilization

text: towards

text: unprecedented

text: levels

text: of

text: prosperity

text: and,

text: by

text: proxy,

text: greenhouse

text: gas

text: emissions.

text: This

text: paper

text: examines

text: whether

text: the

text: theoretical

text: model

text: introduced

text: in

text: Part

text: can

text: explain

text: the

text: evolution

text: of

text: this

text: front.

text: Section

text: outlines

text: the

text: philosophical

text: and

text: thermodynamic

text: basis

text: for

text: describing

text: economic

text: evolution

text: with

text: physics.

text: Section

text: evaluates

text: this

text: model

text: from

text: hindcasts.

text: Sections

text: and

text: discuss

text: and

text: summarize

text: the

text: results.

text: Forces

text: for

text: economic

text: growth

text: The

text: relationship

text: of

text: energy

text: dissipation

text: to

text: human

text: wealth

text: A

text: formal

text: framework

text: for

text: the

text: non-equilibrium

text: thermodynamics

text: of

text: civilization

text: growth

text: was

text: laid

text: out

text: in

text: Part

text: (Garrett,

text: The

text: basis

text: for

text: a

text: model

text: linking

text: economics

text: to

text: physics

text: is

text: a

text: fundamental

text: identity

text: that

text: relates

text: a

text: monetary

text: expression

text: for

text: wealth

text: to

text: the

text: rate

text: at

text: which

text: civilization

text: powers

text: itself

text: with

text: primary

text: energy

text: sources

text: (Garrett,

text: We

text: all

text: have

text: some

text: sense

text: of

text: the

text: difference

text: between

text: civilization

text: and

text: its

text: uncivilized

text: surroundings

text: since

text: we

text: have

text: farms,

text: buildings,

text: human

text: population,

text: vehicles,

text: and

text: communication

text: networks.

text: As

text: shown

text: in

text: Fig.

text: and

text: discussed

text: in

text: Appendix

text: A,

text: this

text: difference

text: implies

text: the

text: existence

text: of

text: a

text: gradient

text: between

text: civilization

text: and

text: its

text: environment.

text: Gradients

text: allow

text: for

text: irreversible

text: thermodynamic

text: ﬂows.

text: A

text: consumption

text: and

text: dissipation

text: of

text: potential

text: energy

text: by

text: civilization

text: sustains

text: internal

text: reversible

text: circulations

text: within

text: civilization

text: that

text: characterize

text: all

text: its

text: activities.

text: The

text: hypothesis

text: that

text: was

text: made

text: in

text: Part

text: is

text: that

text: civilization

text: is

text: effectively

text: a

text: heat

text: engine

text: whose

text: power

text: can

text: be

text: rep-

text: resented

text: in

text: more

text: human

text: terms

text: as

text: economic

text: wealth.

text: Absent

text: any

text: energy

text: consumption,

text: civilization

text: would

text: necessarily

text: decay

text: towards

text: an

text: uncivilized,

text: worthless

text: equilibrium

text: where

text: the

text: gradient

text: ceased

text: to

text: exist

text: and

text: all

text: internal

text: circulations

text: stopped.

text: Wealth

text: is

text: able

text: to

text: grow

text: only

text: when

text: net

text: work

text: is

text: done

text: to

text: grow

text: into

text: the

text: environment.

text: Real

text: economic

text: production

text: occurs

text: when

text: raw

text: materials

text: can

text: be

text: incorporated

text: into

text: civilization’s

text: structure

text: faster

text: than

text: civilization

text: decays

text: (Fig.

text: Growth

text: at

text: a

text: net

text: positive

text: rate

text: expands

text: civilization’s

text: interface

text: with

text: reserves

text: of

text: energy.

text: It

text: enlarges

text: civilization,

text: creating

text: new

text: wealth

text: and

text: a

text: greater

text: overall

text: capacity

text: of

text: civilization

text: to

text: consume

text: and

text: sustain

text: internal

text: circulations.

text: Expressed

text: as

text: an

text: integral,

text: current

text: economic

text: wealth

text: is

text: the

text: net

text: accumulation

text: of

text: past

text: net

text: physical

text: work

text: and

text: real

text: economic

text: production.

text: In

text: effect,

text: there

text: is

text: no

text: intrinsic

text: wealth

text: in

text: and

text: of

text: itself.

text: Rather,

text: as

text: discussed

text: in

text: Part

text: wealth

text: is

text: built

text: from

text: a

text: network


text: Here, GWP growth rates also saturate at a value of ηtech/2, but if G > 1 then this is by way of decline rather than growth. Thus, rates of return on wealth (Eq. 18) and rates of GWP growth (Eq. 20) should have a tendency to converge with time. This is in fact precisely the behavior that has been observed in the past few decades. Figure 4 shows values of η and dlnY/dt that differ by about a factor of 4 in 1950 but that are approaching from opposite directions towards a common value of about 2.5 % per year. 3 Model validation through hindcasts Equation (11) is a new expression for the long-run evolution of the global economy and its resource consumption. Three approaches are now taken to test its validity. 3.1 The functional form relating innovation to growth Figure 6 shows the relationship between innovation rates and rates of return over the past three centuries derived from GWP estimates from Maddison (2003) and the United Nations (Nations, 2010), using Eqs. (6) and (7) (see Sect. 2 and the Supporting Information of Part 1 for methods and associated statistics). Rapid innovation and accelerating rates of return characterized the industrial revolution and the late 1940s. Periods of subsiding innovation followed 1910 and 1950. Equation (11) implies that, if rates of technological −change ηtech are roughly a constant, innovation rates dlnη/dt should be related to rates of return η by a slope of about 2; the intercept should be equivalent to the rate of technolog- ical change η given by Eq. (13). Focussing on the period since 1950, where statistical reconstructions of GWP are tech yearly and presumably most reliable (Maddison, 2003), Fig. 6 shows that the past 60 years have been characterized by a least-squares ﬁt relationship between innovation rates dlnη/dt and rates of return η (with 95 % uncertainty bounds) given by dlnη − ± ± = (2.54 0.54)η +(0.06 0.01). (21) dt Within the stated uncertainty, the observed slope relating innovation to rates of return is consistent with the theoret- − ically expected value of 2 that comes from a law of diminishing returns. The implied rate of technological discovery for this time period η is the intercept of the ﬁt, or about 6 % per year. The magnitude of the difference of the ﬁt tech from the anticipated slope might be an indication that η has not in fact been a strict constant but rather has declined tech with slowly with time, as discussed below. 3.2 Hindcasts of long-run civilization growth The second test is to approach the problem as a hindcast. A hypothetical economic forecaster in 1960 might have noted that the average values of η and dlnη/dt between 1950 and 1960 were 0.9 % and 3.3 % per year, respectively. From Eq. (11), this implies that η was 5.1 % per year during this period. Applying Eqs. (18) and (20), and using tech an initial value for η of 1.0 % per year in 1960, the forecaster could then have obtained the trajectories for economic 0 innovation and growth that are shown in Fig. 7. Fifty-year hindcasts are summarized in Table 1 along with skill scores deﬁned relative to a reference model of persistence in trends (Society, 2014): | | − error(hindcast) Skill score = 1 | |. (22) error(persistence) Skill scores are positive when the hindcast beats persistence in trends, and zero when they do not. For example, average rates of energy consumption growth in the past decade would have been forecast to be 2.3 % per year relative to an observed average of 2.4 % per year. Relative to a persistence prediction of 1.0 % per year, the skill score is 96 %. Alternatively, a forecast of the GWP growth rate for the ﬁrst decade of this century would have been 2.8 % per year compared to the actual observed rate of 2.6 % per year. The persistence forecast based on the 1950 to 1960 period is 4.0 % per year, so the skill score is 91 %. 3.3 Observed magnitude of technological change 11 ##LTLine## ##LTLine##

text: Here,

label: paragraph

text: GWP

text: growth

text: rates

text: also

text: saturate

text: at

text: a

text: value

text: of

text: but

text: if

text: G

text: then

text: this

text: is

text: by

text: way

text: of

text: decline

text: rather

text: than

text: growth.

text: Thus,

text: rates

text: of

text: return

text: on

text: wealth

text: (Eq.

text: and

text: rates

text: of

text: GWP

text: growth

text: (Eq.

text: should

text: have

text: a

text: tendency

text: to

text: converge

text: with

text: time.

text: This

text: is

text: in

text: fact

text: precisely

text: the

text: behavior

text: that

text: has

text: been

text: observed

text: in

text: the

text: past

text: few

text: decades.

text: Figure

text: shows

text: values

text: of

text: η

text: and

text: dlnY/dt

text: that

text: differ

text: by

text: about

text: a

text: factor

text: of

text: in

text: but

text: that

text: are

text: approaching

text: from

text: opposite

text: directions

text: towards

text: a

text: common

text: value

text: of

text: about

text: per

text: year.

text: Model

text: validation

text: through

text: hindcasts

text: Equation

text: is

text: a

text: new

text: expression

text: for

text: the

text: long-run

text: evolution

text: of

text: the

text: global

text: economy

text: and

text: its

text: resource

text: consumption.

text: Three

text: approaches

text: are

text: now

text: taken

text: to

text: test

text: its

text: validity.

text: The

text: functional

text: form

text: relating

text: innovation

text: to

text: growth

text: Figure

text: shows

text: the

text: relationship

text: between

text: innovation

text: rates

text: and

text: rates

text: of

text: return

text: over

text: the

text: past

text: three

text: centuries

text: derived

text: from

text: GWP

text: estimates

text: from

text: Maddison

text: and

text: the

text: United

text: Nations

text: (Nations,

text: using

text: Eqs.

text: and

text: (see

text: Sect.

text: and

text: the

text: Supporting

text: Information

text: of

text: Part

text: for

text: methods

text: and

text: associated

text: statistics).

text: Rapid

text: innovation

text: and

text: accelerating

text: rates

text: of

text: return

text: characterized

text: the

text: industrial

text: revolution

text: and

text: the

text: late

text: Periods

text: of

text: subsiding

text: innovation

text: followed

text: and

text: Equation

text: implies

text: that,

text: if

text: rates

text: of

text: technological

text: −change

text: ηtech

text: are

text: roughly

text: a

text: constant,

text: innovation

text: rates

text: dlnη/dt

text: should

text: be

text: related

text: to

text: rates

text: of

text: return

text: η

text: by

text: a

text: slope

text: of

text: about

text: the

text: intercept

text: should

text: be

text: equivalent

text: to

text: the

text: rate

text: of

text: technolog-

text: ical

text: change

text: η

text: given

text: by

text: Eq.

text: Focussing

text: on

text: the

text: period

text: since

text: where

text: statistical

text: reconstructions

text: of

text: GWP

text: are

text: tech

text: yearly

text: and

text: presumably

text: most

text: reliable

text: (Maddison,

text: Fig.

text: shows

text: that

text: the

text: past

text: years

text: have

text: been

text: characterized

text: by

text: a

text: least-squares

text: ﬁt

text: relationship

text: between

text: innovation

text: rates

text: dlnη/dt

text: and

text: rates

text: of

text: return

text: η

text: (with

text: uncertainty

text: bounds)

text: given

text: by

text: dlnη

text: dt

text: Within

text: the

text: stated

text: uncertainty,

text: the

text: observed

text: slope

text: relating

text: innovation

text: to

text: rates

text: of

text: return

text: is

text: consistent

text: with

text: the

text: theoret-

text: ically

text: expected

text: value

text: of

text: that

text: comes

text: from

text: a

text: law

text: of

text: diminishing

text: returns.

text: The

text: implied

text: rate

text: of

text: technological

text: discovery

text: for

text: this

text: time

text: period

text: η

text: is

text: the

text: intercept

text: of

text: the

text: ﬁt,

text: or

text: about

text: per

text: year.

text: The

text: magnitude

text: of

text: the

text: difference

text: of

text: the

text: ﬁt

text: tech

text: from

text: the

text: anticipated

text: slope

text: might

text: be

text: an

text: indication

text: that

text: η

text: has

text: not

text: in

text: fact

text: been

text: a

text: strict

text: constant

text: but

text: rather

text: has

text: declined

text: tech

text: with

text: slowly

text: with

text: time,

text: as

text: discussed

text: below.

text: Hindcasts

text: of

text: long-run

text: civilization

text: growth

text: The

text: second

text: test

text: is

text: to

text: approach

text: the

text: problem

text: as

text: a

text: hindcast.

text: A

text: hypothetical

text: economic

text: forecaster

text: in

text: might

text: have

text: noted

text: that

text: the

text: average

text: values

text: of

text: η

text: and

text: dlnη/dt

text: between

text: and

text: were

text: and

text: per

text: year,

text: respectively.

text: From

text: Eq.

text: this

text: implies

text: that

text: η

text: was

text: per

text: year

text: during

text: this

text: period.

text: Applying

text: Eqs.

text: and

text: and

text: using

text: tech

text: an

text: initial

text: value

text: for

text: η

text: of

text: per

text: year

text: in

text: the

text: forecaster

text: could

text: then

text: have

text: obtained

text: the

text: trajectories

text: for

text: economic

text: innovation

text: and

text: growth

text: that

text: are

text: shown

text: in

text: Fig.

text: Fifty-year

text: hindcasts

text: are

text: summarized

text: in

text: Table

text: along

text: with

text: skill

text: scores

text: deﬁned

text: relative

text: to

text: a

text: reference

text: model

text: of

text: persistence

text: in

text: trends

text: (Society,

text: error(hindcast)

text: Skill

text: score

text: error(persistence)

text: Skill

text: scores

text: are

text: positive

text: when

text: the

text: hindcast

text: beats

text: persistence

text: in

text: trends,

text: and

text: zero

text: when

text: they

text: do

text: not.

text: For

text: example,

text: average

text: rates

text: of

text: energy

text: consumption

text: growth

text: in

text: the

text: past

text: decade

text: would

text: have

text: been

text: forecast

text: to

text: be

text: per

text: year

text: relative

text: to

text: an

text: observed

text: average

text: of

text: per

text: year.

text: Relative

text: to

text: a

text: persistence

text: prediction

text: of

text: per

text: year,

text: the

text: skill

text: score

text: is

text: Alternatively,

text: a

text: forecast

text: of

text: the

text: GWP

text: growth

text: rate

text: for

text: the

text: ﬁrst

text: decade

text: of

text: this

text: century

text: would

text: have

text: been

text: per

text: year

text: compared

text: to

text: the

text: actual

text: observed

text: rate

text: of

text: per

text: year.

text: The

text: persistence

text: forecast

text: based

text: on

text: the

text: to

text: period

text: is

text: per

text: year,

text: so

text: the

text: skill

text: score

text: is

text: Observed

text: magnitude

text: of

text: technological

text: change

text: ##LTLine##

text: ##LTLine##


text: dK − − − = (Y W) δK = sY δK, (23) dt − where individual and government consumption is represented by W = (1 s)Y. What is not saved or invested in the future is consumed in the present. Labor L (in units of worker hours) uses accumulated investments in physical capital to enable further production Y according to some functional form f(K, L). A commonly used representation is the Cobb–Douglas production function − Y = AKαL1 α, (24) where A is a “total factor productivity” that accounts for any residual in the output Y that is not explained by the (cid:54) inputs K and L. The exponent α is empirically determined from a ﬁt to past data and α = 1. Unfortunately, this presents the drawback that the units for A are ill-deﬁned and dependent on the scenario considered. The Solow growth model (Solow, 1957) expresses the prognostic form for Eq. (24) as dlnY dlnA dlnK − dlnL = +α +(1 α) . (25) dt dt dt dt The term dlnA/dt has often been interpreted to represent technological progress. Such progress might be exoge- nous (Solow, 1957) or endogenous (Grossman and Helpman, 1990; Romer, 1994). If exogenous, then progress is considered to be due to an unknown external force. If endogenous, then it might come from targeted investments such as research and development. In comparison, the alternative approach that has been presented here considers civilization as a whole. Labor is subsumed into total capital. The physics of energy dissipation suggest a focus on the connections between compo- nents within a global network rather than on the elements themselves. This adjustment requires only a slight, though important, modiﬁcation to the Solow growth model in which α = 1 and Y = AK, where A has ﬁxed units of inverse time. In this case, no ﬁt to data is required to obtain α, and the units for A are consistent and physical. Equation (25) ≡ ≡ becomes equivalent to the expression Y = ηC in Eq. (5), where η A and C K. Further, the expression dlnA/dt that is assumed to describe technological progress in neoclassical frameworks (Eq. 25) is then mathematically equivalent to the deﬁnition for innovation dlnη/dt in the thermodynamic framework (Eq. 7). In energy economics, the term “production efﬁciency”, or its inverse, the “energy intensity”, is often used to relate the amount of economic output that society is able to obtain per unit of energy it consumes (Sorrell, 2007). More efﬁcient, less energy-intense production is ascribed to technological change (e.g., Pielke Jr et al., 2008). The production efﬁciency can be deﬁned mathematically as the ratio f = Y/a. (26) From Eq. (26) and the expression a = λC (Eq. 1), where λ is a constant, the production efﬁciency can then be linked to wealth C through 1 Y f = . (27) λ C With rearrangement, λ f is then equivalent to the rate of return η in Eq. (5) that expresses how fast economic wealth C can be converted to economic production Y through Y = ηC. More efﬁcient production leads to faster growth of wealth through 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dK

label: equation

text: (Y

text: W)

text: δK

text: sY

text: δK,

text: dt

text: where

text: individual

text: and

text: government

text: consumption

text: is

text: represented

text: by

text: W

text: s)Y.

text: What

text: is

text: not

text: saved

text: or

text: invested

text: in

text: the

text: future

text: is

text: consumed

text: in

text: the

text: present.

text: Labor

text: L

text: (in

text: units

text: of

text: worker

text: hours)

text: uses

text: accumulated

text: investments

text: in

text: physical

text: capital

text: to

text: enable

text: further

text: production

text: Y

text: according

text: to

text: some

text: functional

text: form

text: f(K,

text: L).

text: A

text: commonly

text: used

text: representation

text: is

text: the

text: Cobb–Douglas

text: production

text: function

text: Y

text: α,

text: where

text: A

text: is

text: a

text: “total

text: factor

text: productivity”

text: that

text: accounts

text: for

text: any

text: residual

text: in

text: the

text: output

text: Y

text: that

text: is

text: not

text: explained

text: by

text: the

text: inputs

text: K

text: and

text: L.

text: The

text: exponent

text: α

text: is

text: empirically

text: determined

text: from

text: a

text: ﬁt

text: to

text: past

text: data

text: and

text: α

text: Unfortunately,

text: this

text: presents

text: the

text: drawback

text: that

text: the

text: units

text: for

text: A

text: are

text: ill-deﬁned

text: and

text: dependent

text: on

text: the

text: scenario

text: considered.

text: The

text: Solow

text: growth

text: model

text: (Solow,

text: expresses

text: the

text: prognostic

text: form

text: for

text: Eq.

text: as

text: dlnY

text: dlnA

text: dlnK

text: dlnL

text: +α

text: α)

text: dt

text: dt

text: dt

text: dt

text: The

text: term

text: dlnA/dt

text: has

text: often

text: been

text: interpreted

text: to

text: represent

text: technological

text: progress.

text: Such

text: progress

text: might

text: be

text: exoge-

text: nous

text: (Solow,

text: or

text: endogenous

text: (Grossman

text: and

text: Helpman,

text: Romer,

text: If

text: exogenous,

text: then

text: progress

text: is

text: considered

text: to

text: be

text: due

text: to

text: an

text: unknown

text: external

text: force.

text: If

text: endogenous,

text: then

text: it

text: might

text: come

text: from

text: targeted

text: investments

text: such

text: as

text: research

text: and

text: development.

text: In

text: comparison,

text: the

text: alternative

text: approach

text: that

text: has

text: been

text: presented

text: here

text: considers

text: civilization

text: as

text: a

text: whole.

text: Labor

text: is

text: subsumed

text: into

text: total

text: capital.

text: The

text: physics

text: of

text: energy

text: dissipation

text: suggest

text: a

text: focus

text: on

text: the

text: connections

text: between

text: compo-

text: nents

text: within

text: a

text: global

text: network

text: rather

text: than

text: on

text: the

text: elements

text: themselves.

text: This

text: adjustment

text: requires

text: only

text: a

text: slight,

text: though

text: important,

text: modiﬁcation

text: to

text: the

text: Solow

text: growth

text: model

text: in

text: which

text: α

text: and

text: Y

text: AK,

text: where

text: A

text: has

text: ﬁxed

text: units

text: of

text: inverse

text: time.

text: In

text: this

text: case,

text: no

text: ﬁt

text: to

text: data

text: is

text: required

text: to

text: obtain

text: α,

text: and

text: the

text: units

text: for

text: A

text: are

text: consistent

text: and

text: physical.

text: Equation

text: becomes

text: equivalent

text: to

text: the

text: expression

text: Y

text: ηC

text: in

text: Eq.

text: where

text: η

text: A

text: and

text: C

text: K.

text: Further,

text: the

text: expression

text: dlnA/dt

text: that

text: is

text: assumed

text: to

text: describe

text: technological

text: progress

text: in

text: neoclassical

text: frameworks

text: (Eq.

text: is

text: then

text: mathematically

text: equivalent

text: to

text: the

text: deﬁnition

text: for

text: innovation

text: dlnη/dt

text: in

text: the

text: thermodynamic

text: framework

text: (Eq.

text: In

text: energy

text: economics,

text: the

text: term

text: “production

text: efﬁciency”,

text: or

text: its

text: inverse,

text: the

text: “energy

text: intensity”,

text: is

text: often

text: used

text: to

text: relate

text: the

text: amount

text: of

text: economic

text: output

text: that

text: society

text: is

text: able

text: to

text: obtain

text: per

text: unit

text: of

text: energy

text: it

text: consumes

text: (Sorrell,

text: More

text: efﬁcient,

text: less

text: energy-intense

text: production

text: is

text: ascribed

text: to

text: technological

text: change

text: (e.g.,

text: Pielke

text: Jr

text: et

text: al.,

text: The

text: production

text: efﬁciency

text: can

text: be

text: deﬁned

text: mathematically

text: as

text: the

text: ratio

text: f

text: Y/a.

text: From

text: Eq.

text: and

text: the

text: expression

text: a

text: λC

text: (Eq.

text: where

text: λ

text: is

text: a

text: constant,

text: the

text: production

text: efﬁciency

text: can

text: then

text: be

text: linked

text: to

text: wealth

text: C

text: through

text: Y

text: f

text: λ

text: C

text: With

text: rearrangement,

text: λ

text: f

text: is

text: then

text: equivalent

text: to

text: the

text: rate

text: of

text: return

text: η

text: in

text: Eq.

text: that

text: expresses

text: how

text: fast

text: economic

text: wealth

text: C

text: can

text: be

text: converted

text: to

text: economic

text: production

text: Y

text: through

text: Y

text: ηC.

text: More

text: efﬁcient

text: production

text: leads

text: to

text: faster

text: growth

text: of

text: wealth

text: through

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Ku¨mmel, R. (2011), The second law of economics, Springer. Lennie, P. (2003), The cost of cortical computation, Current Biology, 13(6), 493 – 497, doi:10.1016/S0960- 9822(03)00135-0. Lotka, A. J. (1922), Contribution to the energetics of evolution, Proc. Nat. Acad. Sci., 8, 147–151. Maddison, A. (2003), The World Economy: Historical Statistics, 276 pp., OECD, (last access: April 2010). Marchetti, C., and J. H. Ausubel (2012), Quantitative dynamics of human empires, Int. J. Anth., 27, 1–62. Matthews, H. D., N. P. Gillett, P. A. Stott, and K. Zickfeld (2009), The proportionality of global warming to cumulative carbon emissions, Nature, 459(7248), 829–832, doi:10.1038/nature08047. Mora, C., et al. (2013), The projected timing of climate departure from recent variability, Nature, 502(7470), 183–187, doi:10.1038/nature12540. Moss, R. H., et al. (2010), The next generation of scenarios for climate change research and assessment, Nature, 463(7282), 747–756, doi:10.1038/nature08823. Murphy, D. J., and C. A. S. Hall (2010), Year in review–EROI or energy return on (energy) invested, Ann. New York Acad. Sci., 1185(1), 102–118, doi:10.1111/j.1749-6632.2009.05282.x. Murray, J., and D. King (2012), Climate policy: Oil’s tipping point has passed, Nature, 481, 433–435, doi: 10.1038/481433a. Nations, U. (2010), United Nations Statistical Databases, (last access: June 2014). Nordhaus, W., and P. Sztorc (2013), DICE 2013R: Introduction and User’s Manual. Nordhaus, W. D. (2010), Economic aspects of global warming in a post-Copenhagen environment, Proc. Nat. Acad. Sci., 107, 11,721–11,726, doi:10.1073/pnas.1005985107. Odum, H. T. (1971), Environment, power, and society, vol. 130, Wiley-Interscience, New York. Pacala, S., and R. Socolow (2004), Stabilization wedges: Solving the climate problem for the next 50 years with current technologies, Science, 305, 968–972, doi:10.1126/science.1100103. Peters, G. P., R. M. Andrew, T. Boden, J. G. Canadell, P. Ciais, C. Le Quere, G. Marland, M. R. Raupach, and ◦ C. Wilson (2013), The challenge to keep global warming below 2 C, Nature Clim. Change, 3(1), 4–6, doi: 10.1038/nclimate1783. Pielke Jr, R., T. Wigley, and C. Green (2008), Dangerous assumptions, Nature, 452, 531–532. Pindyck, R. S. (2013), Climate change policy: What do the models tell us?, J. Econ. Lit., 51, 860–872, doi: 10.1257/jel.51.3.860. Pruppacher, H. R., and J. D. Klett (1997), Microphysics of Clouds and Precipitation, 2nd Rev. Edn., Kluwer Academic Publishing, Dordrecht. Raupach, M. R., G. Marland, P. Ciais, C. Le Que´re´, J. G. Canadell, G. Klepper, and C. Field (2007), Global and regional drivers of accelerating CO emissions, Proc. Nat. Acad. Sci., 104, 10,288–10,293, doi: 2 10.1073/pnas.0700609104. Rogers, E. M. (2010), Diffusion of innovations, Simon and Schuster. Romer, P. M. (1994), The origins of endogenous growth, J. Econ. Perspect., 8, 3–22. Rutledge, D. (2011), Estimating long-term world coal production with logit and probit transforms, International Jour- nal of Coal Geology, 85(1), 23 – 33, doi:10.1016/j.coal.2010.10.012. 21

text: Ku¨mmel,

label: reference

text: R.

text: The

text: second

text: law

text: of

text: economics,

text: Springer.

text: Lennie,

text: P.

text: The

text: cost

text: of

text: cortical

text: computation,

text: Current

text: Biology,

text: Lotka,

text: A.

text: J.

text: Contribution

text: to

text: the

text: energetics

text: of

text: evolution,

text: Proc.

text: Nat.

text: Acad.

text: Sci.,

text: Maddison,

text: A.

text: The

text: World

text: Economy:

text: Historical

text: Statistics,

text: pp.,

text: OECD,

text: (last

text: access:

text: April

text: Marchetti,

text: C.,

text: and

text: J.

text: H.

text: Ausubel

text: Quantitative

text: dynamics

text: of

text: human

text: empires,

text: Int.

text: J.

text: Anth.,

text: Matthews,

text: H.

text: D.,

text: N.

text: P.

text: Gillett,

text: P.

text: A.

text: Stott,

text: and

text: K.

text: Zickfeld

text: The

text: proportionality

text: of

text: global

text: warming

text: to

text: cumulative

text: carbon

text: emissions,

text: Nature,

text: Mora,

text: C.,

text: et

text: al.

text: The

text: projected

text: timing

text: of

text: climate

text: departure

text: from

text: recent

text: variability,

text: Nature,

text: Moss,

text: R.

text: H.,

text: et

text: al.

text: The

text: next

text: generation

text: of

text: scenarios

text: for

text: climate

text: change

text: research

text: and

text: assessment,

text: Nature,

text: Murphy,

text: D.

text: J.,

text: and

text: C.

text: A.

text: S.

text: Hall

text: Year

text: in

text: review–EROI

text: or

text: energy

text: return

text: on

text: (energy)

text: invested,

text: Ann.

text: New

text: York

text: Acad.

text: Sci.,

text: Murray,

text: J.,

text: and

text: D.

text: King

text: Climate

text: policy:

text: Oil’s

text: tipping

text: point

text: has

text: passed,

text: Nature,

text: doi:

text: Nations,

text: U.

text: United

text: Nations

text: Statistical

text: Databases,

text: (last

text: access:

text: June

text: Nordhaus,

text: W.,

text: and

text: P.

text: Sztorc

text: DICE

text: Introduction

text: and

text: User’s

text: Manual.

text: Nordhaus,

text: W.

text: D.

text: Economic

text: aspects

text: of

text: global

text: warming

text: in

text: a

text: post-Copenhagen

text: environment,

text: Proc.

text: Nat.

text: Acad.

text: Sci.,

text: Odum,

text: H.

text: T.

text: Environment,

text: power,

text: and

text: society,

text: vol.

text: Wiley-Interscience,

text: New

text: York.

text: Pacala,

text: S.,

text: and

text: R.

text: Socolow

text: Stabilization

text: wedges:

text: Solving

text: the

text: climate

text: problem

text: for

text: the

text: next

text: years

text: with

text: current

text: technologies,

text: Science,

text: Peters,

text: G.

text: P.,

text: R.

text: M.

text: Andrew,

text: T.

text: Boden,

text: J.

text: G.

text: Canadell,

text: P.

text: Ciais,

text: C.

text: Le

text: Quere,

text: G.

text: Marland,

text: M.

text: R.

text: Raupach,

text: and

text: C.

text: Wilson

text: The

text: challenge

text: to

text: keep

text: global

text: warming

text: below

text: C,

text: Nature

text: Clim.

text: Change,

text: doi:

text: Pielke

text: Jr,

text: R.,

text: T.

text: Wigley,

text: and

text: C.

text: Green

text: Dangerous

text: assumptions,

text: Nature,

text: Pindyck,

text: R.

text: S.

text: Climate

text: change

text: policy:

text: What

text: do

text: the

text: models

text: tell

text: us?,

text: J.

text: Econ.

text: Lit.,

text: doi:

text: Pruppacher,

text: H.

text: R.,

text: and

text: J.

text: D.

text: Klett

text: Microphysics

text: of

text: Clouds

text: and

text: Precipitation,

text: Rev.

text: Edn.,

text: Kluwer

text: Academic

text: Publishing,

text: Dordrecht.

text: Raupach,

text: M.

text: R.,

text: G.

text: Marland,

text: P.

text: Ciais,

text: C.

text: Le

text: Que´re´,

text: J.

text: G.

text: Canadell,

text: G.

text: Klepper,

text: and

text: C.

text: Field

text: Global

text: and

text: regional

text: drivers

text: of

text: accelerating

text: CO

text: emissions,

text: Proc.

text: Nat.

text: Acad.

text: Sci.,

text: doi:

text: Rogers,

text: E.

text: M.

text: Diffusion

text: of

text: innovations,

text: Simon

text: and

text: Schuster.

text: Romer,

text: P.

text: M.

text: The

text: origins

text: of

text: endogenous

text: growth,

text: J.

text: Econ.

text: Perspect.,

text: Rutledge,

text: D.

text: Estimating

text: long-term

text: world

text: coal

text: production

text: with

text: logit

text: and

text: probit

text: transforms,

text: International

text: Jour-

text: nal

text: of

text: Coal

text: Geology,


text: 1.90 %/yr 1 1.90 %/yr ! t Y dt! (1015 $) 0 a (1021 J/yr ) λ = a/! t Y dt! (10− 2 W/$) 0 0.1 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year (cid:82) t (cid:48) (cid:48) Figure 3: Rates of global energy consumption a, global wealth C = Y(t ) dt , and the ratio λ = a/C since 1970. 0 The average rates of growth η for a and C in percent per year are shown for comparison. The average value of λ is ± 7.1 0.1 mW per year 2005 USD. Note the y axis is a logarithmic scale. 20 % of the daily caloric input to the body as a whole (Varela et al., 2001; Lennie, 2003; Buzsa´ki and Draguhn, 2004). Perhaps dissipative neuronal circulations along brain networks reﬂect our collective perception of real global economic wealth. They march to broader economic circulations along global civilization networks that are sustained by a dissipation of oil, coal, and other primary energy supplies. Eqs. (1) to (3) may seem unorthodox by traditional economic standards, but there may be some basis for interpreting λ as a type of psychological constant that links the physics of human perception to the thermodynamic ﬂows that drive the global economy. As a point of comparison, traditional economic growth models are divorced from expressions of energy dissipation and physical ﬂows, where wealth is expressed in terms of a physical capital, or as a stock that has an intrinsic value. New capital is produced using currently existing labor and capital, and production levels have no explicit dependence on external physical constraints (Solow, 1956). A more detailed outline and juxtaposition with the model here is described in Appendix B. The ﬁeld of macroeconomics is however making steps towards creating links with physics, pointing out that, along with labor and capital, energy must also be a factor of economic production (Lotka, 1922; Soddy, 1933; Odum, 1971; Georgescu-Roegen, 1993; Hua and Bakshi, 2004; Annila and Salthe, 2009). Quantiﬁed links between physical and ﬁnancial quantities often rely upon a high observed correlation between national or sectoral economic production and energy consumption (Costanza, 1980; Cleveland et al., 1984; Brown et al., 2011). A few economic growth models use these data to partially substitute energy for labor and capital as a motive productive force (Ayres and Warr, 2009; Ku¨mmel, 2011). The model presented here differs, foremost because civilization is examined only as a whole, as an evolving organism whose growth is a response only to its changing ability to access external resources (Gowdy and Krall, 2013; Herrmann-Pillath, 2015). Nothing is said about internal trade between countries. Neither is a distinction made 5 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: %/yr

text: %/yr

text: t

text: Y

text: dt!

text: a

text: J/yr

text: λ

text: a/!

text: t

text: Y

text: dt!

text: W/$)

text: Year

label: caption

text: t

text: Figure

text: Rates

text: of

text: global

text: energy

text: consumption

text: a,

text: global

text: wealth

text: C

text: Y(t

text: dt

text: and

text: the

text: ratio

text: λ

text: a/C

text: since

text: The

text: average

text: rates

text: of

text: growth

text: η

text: for

text: a

text: and

text: C

text: in

text: percent

text: per

text: year

text: are

text: shown

text: for

text: comparison.

text: The

text: average

text: value

text: of

text: λ

text: is

text: mW

text: per

text: year

text: USD.

text: Note

text: the

text: y

text: axis

text: is

text: a

text: logarithmic

text: scale.

text: of

text: the

text: daily

text: caloric

text: input

text: to

text: the

text: body

text: as

text: a

text: whole

text: (Varela

text: et

text: al.,

text: Lennie,

text: Buzsa´ki

text: and

text: Draguhn,

text: Perhaps

text: dissipative

text: neuronal

text: circulations

text: along

text: brain

text: networks

text: reﬂect

text: our

text: collective

text: perception

text: of

text: real

text: global

text: economic

text: wealth.

text: They

text: march

text: to

text: broader

text: economic

text: circulations

text: along

text: global

text: civilization

text: networks

text: that

text: are

text: sustained

text: by

text: a

text: dissipation

text: of

text: oil,

text: coal,

text: and

text: other

text: primary

text: energy

text: supplies.

text: Eqs.

text: to

text: may

text: seem

text: unorthodox

text: by

text: traditional

text: economic

text: standards,

text: but

text: there

text: may

text: be

text: some

text: basis

text: for

text: interpreting

text: λ

text: as

text: a

text: type

text: of

text: psychological

text: constant

text: that

text: links

text: the

text: physics

text: of

text: human

text: perception

text: to

text: the

text: thermodynamic

text: ﬂows

text: that

text: drive

text: the

text: global

text: economy.

text: As

text: a

text: point

text: of

text: comparison,

text: traditional

text: economic

text: growth

text: models

text: are

text: divorced

text: from

text: expressions

text: of

text: energy

text: dissipation

text: and

text: physical

text: ﬂows,

text: where

text: wealth

text: is

text: expressed

text: in

text: terms

text: of

text: a

text: physical

text: capital,

text: or

text: as

text: a

text: stock

text: that

text: has

text: an

text: intrinsic

text: value.

text: New

text: capital

text: is

text: produced

text: using

text: currently

text: existing

text: labor

text: and

text: capital,

text: and

text: production

text: levels

text: have

text: no

text: explicit

text: dependence

text: on

text: external

text: physical

text: constraints

text: (Solow,

text: A

text: more

text: detailed

text: outline

text: and

text: juxtaposition

text: with

text: the

text: model

text: here

text: is

text: described

text: in

text: Appendix

text: B.

text: The

text: ﬁeld

text: of

text: macroeconomics

text: is

text: however

text: making

text: steps

text: towards

text: creating

text: links

text: with

text: physics,

text: pointing

text: out

text: that,

text: along

text: with

text: labor

text: and

text: capital,

text: energy

text: must

text: also

text: be

text: a

text: factor

text: of

text: economic

text: production

text: (Lotka,

text: Soddy,

text: Odum,

text: Georgescu-Roegen,

text: Hua

text: and

text: Bakshi,

text: Annila

text: and

text: Salthe,

text: Quantiﬁed

text: links

text: between

text: physical

text: and

text: ﬁnancial

text: quantities

text: often

text: rely

text: upon

text: a

text: high

text: observed

text: correlation

text: between

text: national

text: or

text: sectoral

text: economic

text: production

text: and

text: energy

text: consumption

text: (Costanza,

text: Cleveland

text: et

text: al.,

text: Brown

text: et

text: al.,

text: A

text: few

text: economic

text: growth

text: models

text: use

text: these

text: data

text: to

text: partially

text: substitute

text: energy

text: for

text: labor

text: and

text: capital

text: as

text: a

text: motive

text: productive

text: force

text: (Ayres

text: and

text: Warr,

text: Ku¨mmel,

text: The

text: model

text: presented

text: here

text: differs,

text: foremost

text: because

text: civilization

text: is

text: examined

text: only

text: as

text: a

text: whole,

text: as

text: an

text: evolving

text: organism

text: whose

text: growth

text: is

text: a

text: response

text: only

text: to

text: its

text: changing

text: ability

text: to

text: access

text: external

text: resources

text: (Gowdy

text: and

text: Krall,

text: Herrmann-Pillath,

text: Nothing

text: is

text: said

text: about

text: internal

text: trade

text: between

text: countries.

text: Neither

text: is

text: a

text: distinction

text: made

text: ##LTFigure##

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text: Since 1980, they have been closer to 3 % per year. Increasingly, the long-term increase in civilization’s rate of return η has been offset by the long-term decrease in innovation dlnη/dt. The turning point was in the late 1970s, when, as shown in Fig. 4, innovation rates dipped below rates of return. Between 1950 and 1975, current innovation was the largest contributor to current GWP growth rates. Since then, continued GWP growth has relied increasingly on innovations made in the ﬁrst two decades since the end of World War II (Eq. 9). 2.3 Physical forces for innovation For the special case that there is positive net convergence of matter in a system, the system grows (see Fig. 2 and Appendix A). It extends its interface with accessible reserves of energy and matter. An enlarged interface allows for faster rates of consumption. The result is a positive feedback that allows growth to accelerate. This is a basic recipe for emergent or exponential growth. One important aspect of this feedback, however, is that rates of exponential growth are never constant. Rather, they increase when new reserves of energy or matter are discovered and they decrease when there is accelerated decay. The thermodynamics of this recipe (Garrett, 2012a) were applied in Part 1 to the emergent growth of civilization and its rates of return on wealth (Garrett, 2014). It was shown that the rate of return η can be broken down into the proportionality η ∝ (1−δ) ∆HR . (10) N2/3etot S S Here, N represents the amount of matter or mass within civilization. N grows from a positive imbalance between S S civilization’s incorporation of raw materials from the environment and civilization’s material decay. δ is the decay parameter that accounts for how rapidly N falls apart due to natural causes. ∆H represents the size of the energy S R reserves that are available to be consumed by civilization. The term etot represents how much of this energy must be S consumed by civilization in order that raw materials can be added to civilization’s fabric, thereby adding to N . The S exponent 2/3 arises from how ﬂows are down a gradient and across an interface. Building on the identity a = λC, it was argued in Garrett (2014) that Eq. (10) implies that rates of economic innovation can be represented by dlnη − dt = 2η +ηtech (11) (12) − The ﬁrst term represents a drag on innovation due to a law of diminishing returns 2η. The second term expresses a rate of technological change η due to changes in δ, ∆H , and etot. A distinction is made here between a tech- tech R S nological advance and an innovation. Technological change only counts as an innovation if it overcomes diminishing returns to lead to a real increase in the rate of return η. A law of diminishing returns is a characteristic feature of emergent systems. As indicated by Eq. (10), the expo- nential growth rates of larger, older objects with high values of N tend to be lower than for smaller, younger ones. In S our case, our bodies are a complex network of nerves, neurons, veins, gastrointestinal tracts, and pulmonary tubes. We use this network so that we can interact with a network of electrical circuits, communication lines, plumbing, roads, shipping lanes, and aviation routes (van Dijk, 2012). Such networks have been built from a net accumulation of matter. So, as civilization grows, any given addition becomes increasingly incremental. The implication of Eq. (11) is that, absent sufﬁciently rapid technological change, relative growth rates η will tend to decline, and innovation will turn negative. For example, from Eq. (11), inno−vation requires that ηtech > 2η. Or, by substituting Eq. (8) into Eq. (11), an expression for GWP growth is dlnY/dt = η + ηtech, in which case maintenance of positive GWP growth requires that ηtech > η. That economic growth has been sustained over the past 150 years is a testament to the importance of technological change for overcoming diminishing returns. The rate of technological change follows from the ﬁrst derivate of Eq. (10): 8 ##LTLine## ##LTLine##

text: Since

text: they

text: have

text: been

text: closer

text: to

text: per

text: year.

text: Increasingly,

text: the

text: long-term

text: increase

text: in

text: civilization’s

text: rate

text: of

text: return

text: η

text: has

text: been

text: offset

text: by

text: the

text: long-term

text: decrease

text: in

text: innovation

text: dlnη/dt.

text: The

text: turning

text: point

text: was

text: in

text: the

text: late

text: when,

text: as

text: shown

text: in

text: Fig.

text: innovation

text: rates

text: dipped

text: below

text: rates

text: of

text: return.

text: Between

text: and

text: current

text: innovation

text: was

text: the

text: largest

text: contributor

text: to

text: current

text: GWP

text: growth

text: rates.

text: Since

text: then,

text: continued

text: GWP

text: growth

text: has

text: relied

text: increasingly

text: on

text: innovations

text: made

text: in

text: the

text: ﬁrst

text: two

text: decades

text: since

text: the

text: end

text: of

text: World

text: War

text: II

text: (Eq.

label: section

text: Physical

text: forces

text: for

text: innovation

text: For

text: the

text: special

text: case

text: that

text: there

text: is

text: positive

text: net

text: convergence

text: of

text: matter

text: in

text: a

text: system,

text: the

text: system

text: grows

text: (see

text: Fig.

text: and

text: Appendix

text: A).

text: It

text: extends

text: its

text: interface

text: with

text: accessible

text: reserves

text: of

text: energy

text: and

text: matter.

text: An

text: enlarged

text: interface

text: allows

text: for

text: faster

text: rates

text: of

text: consumption.

text: The

text: result

text: is

text: a

text: positive

text: feedback

text: that

text: allows

text: growth

text: to

text: accelerate.

text: This

text: is

text: a

text: basic

text: recipe

text: for

text: emergent

text: or

text: exponential

text: growth.

text: One

text: important

text: aspect

text: of

text: this

text: feedback,

text: however,

text: is

text: that

text: rates

text: of

text: exponential

text: growth

text: are

text: never

text: constant.

text: Rather,

text: they

text: increase

text: when

text: new

text: reserves

text: of

text: energy

text: or

text: matter

text: are

text: discovered

text: and

text: they

text: decrease

text: when

text: there

text: is

text: accelerated

text: decay.

text: The

text: thermodynamics

text: of

text: this

text: recipe

text: (Garrett,

text: were

text: applied

text: in

text: Part

text: to

text: the

text: emergent

text: growth

text: of

text: civilization

text: and

text: its

text: rates

text: of

text: return

text: on

text: wealth

text: (Garrett,

text: It

text: was

text: shown

text: that

text: the

text: rate

text: of

text: return

text: η

text: can

text: be

text: broken

text: down

text: into

text: the

text: proportionality

text: η

text: ∆HR

text: S

text: S

text: Here,

text: N

text: represents

text: the

text: amount

text: of

text: matter

text: or

text: mass

text: within

text: civilization.

text: N

text: grows

text: from

text: a

text: positive

text: imbalance

text: between

text: S

text: S

text: civilization’s

text: incorporation

text: of

text: raw

text: materials

text: from

text: the

text: environment

text: and

text: civilization’s

text: material

text: decay.

text: δ

text: is

text: the

text: decay

text: parameter

text: that

text: accounts

text: for

text: how

text: rapidly

text: N

text: falls

text: apart

text: due

text: to

text: natural

text: causes.

text: ∆H

text: represents

text: the

text: size

text: of

text: the

text: energy

text: S

text: R

text: reserves

text: that

text: are

text: available

text: to

text: be

text: consumed

text: by

text: civilization.

text: The

text: term

text: etot

text: represents

text: how

text: much

text: of

text: this

text: energy

text: must

text: be

text: S

text: consumed

text: by

text: civilization

text: in

text: order

text: that

text: raw

text: materials

text: can

text: be

text: added

text: to

text: civilization’s

text: fabric,

text: thereby

text: adding

text: to

text: N

text: The

text: S

text: exponent

text: arises

text: from

text: how

text: ﬂows

text: are

text: down

text: a

text: gradient

text: and

text: across

text: an

text: interface.

text: Building

text: on

text: the

text: identity

text: a

text: λC,

text: it

text: was

text: argued

text: in

text: Garrett

text: that

text: Eq.

text: implies

text: that

text: rates

text: of

text: economic

text: innovation

text: can

text: be

text: represented

text: by

text: dlnη

text: dt

text: 2η

text: +ηtech

text: The

text: ﬁrst

text: term

text: represents

text: a

text: drag

text: on

text: innovation

text: due

text: to

text: a

text: law

text: of

text: diminishing

text: returns

text: The

text: second

text: term

text: expresses

text: a

text: rate

text: of

text: technological

text: change

text: η

text: due

text: to

text: changes

text: in

text: δ,

text: ∆H

text: and

text: etot.

text: A

text: distinction

text: is

text: made

text: here

text: between

text: a

text: tech-

text: tech

text: R

text: S

text: nological

text: advance

text: and

text: an

text: innovation.

text: Technological

text: change

text: only

text: counts

text: as

text: an

text: innovation

text: if

text: it

text: overcomes

text: diminishing

text: returns

text: to

text: lead

text: to

text: a

text: real

text: increase

text: in

text: the

text: rate

text: of

text: return

text: η.

text: A

text: law

text: of

text: diminishing

text: returns

text: is

text: a

text: characteristic

text: feature

text: of

text: emergent

text: systems.

text: As

text: indicated

text: by

text: Eq.

text: the

text: expo-

text: nential

text: growth

text: rates

text: of

text: larger,

text: older

text: objects

text: with

text: high

text: values

text: of

text: N

text: tend

text: to

text: be

text: lower

text: than

text: for

text: smaller,

text: younger

text: ones.

text: In

text: S

text: our

text: case,

text: our

text: bodies

text: are

text: a

text: complex

text: network

text: of

text: nerves,

text: neurons,

text: veins,

text: gastrointestinal

text: tracts,

text: and

text: pulmonary

text: tubes.

text: We

text: use

text: this

text: network

text: so

text: that

text: we

text: can

text: interact

text: with

text: a

text: network

text: of

text: electrical

text: circuits,

text: communication

text: lines,

text: plumbing,

text: roads,

text: shipping

text: lanes,

text: and

text: aviation

text: routes

text: (van

text: Dijk,

text: Such

text: networks

text: have

text: been

text: built

text: from

text: a

text: net

text: accumulation

text: of

text: matter.

text: So,

text: as

text: civilization

text: grows,

text: any

text: given

text: addition

text: becomes

text: increasingly

text: incremental.

text: The

text: implication

text: of

text: Eq.

text: is

text: that,

text: absent

text: sufﬁciently

text: rapid

text: technological

text: change,

text: relative

text: growth

text: rates

text: η

text: will

text: tend

text: to

text: decline,

text: and

text: innovation

text: will

text: turn

text: negative.

text: For

text: example,

text: from

text: Eq.

text: inno−vation

text: requires

text: that

text: ηtech

text: Or,

text: by

text: substituting

text: Eq.

text: into

text: Eq.

text: an

text: expression

text: for

text: GWP

text: growth

text: is

text: dlnY/dt

text: η

text: ηtech,

text: in

text: which

text: case

text: maintenance

text: of

text: positive

text: GWP

text: growth

text: requires

text: that

text: ηtech

text: η.

text: That

text: economic

text: growth

text: has

text: been

text: sustained

text: over

text: the

text: past

text: years

text: is

text: a

text: testament

text: to

text: the

text: importance

text: of

text: technological

text: change

text: for

text: overcoming

text: diminishing

text: returns.

text: The

text: rate

text: of

text: technological

text: change

text: follows

text: from

text: the

text: ﬁrst

text: derivate

text: of

text: Eq.

text: ##LTLine##

text: ##LTLine##


text: Assessing Time-Varying Causal Eﬀect Moderation in Mobile Health Audrey Boruvka1, Daniel Almirall2, Katie Witkiewitz3, and Susan A. Murphy1,2 1Department of Statistics, University of Michigan 2Institute for Social Research, University of Michigan 3Department of Psychology, University of New Mexico Abstract In mobile health interventions aimed at behavior change and maintenance, treatments are provided in real time to manage current or impending high risk situations or promote healthy behaviors in near real time. Currently there is great scientiﬁc interest in developing data analysis approaches to guide the development of mobile interventions. In particular data from mobile health studies might be used to examine eﬀect moderators— individual characteristics, time-varying context or past treatment response that moderate the eﬀect of current treatment on a subsequent response. This paper introduces a formal deﬁnition for moderated eﬀects in terms of potential outcomes, a deﬁnition that is particularly suited to mobile interventions, where treatment occasions are numerous, individuals are not always available for treatment, and potential moderators might be inﬂuenced by past treatment. Methods for estimating moderated eﬀects are developed and compared. The proposed approach is illustrated using BASICS-Mobile, a smartphone-based intervention designed to curb heavy drinking and smoking among college students. Keywords: mHealth, structural nested mean model, eﬀect modiﬁcation 1

text: Assessing

label: title

text: Time-Varying

text: Causal

text: Eﬀect

text: Moderation

text: in

text: Mobile

text: Health

text: Audrey

label: author

text: Daniel

text: Katie

text: and

text: Susan

text: A.

text: of

text: Statistics,

text: University

text: of

text: Michigan

text: for

text: Social

text: Research,

text: University

text: of

text: Michigan

text: of

text: Psychology,

text: University

text: of

text: New

text: Mexico

text: Abstract

text: In

label: abstract

text: mobile

text: health

text: interventions

text: aimed

text: at

text: behavior

text: change

text: and

text: maintenance,

text: treatments

text: are

text: provided

text: in

text: real

text: time

text: to

text: manage

text: current

text: or

text: impending

text: high

text: risk

text: situations

text: or

text: promote

text: healthy

text: behaviors

text: in

text: near

text: real

text: time.

text: Currently

text: there

text: is

text: great

text: scientiﬁc

text: interest

text: in

text: developing

text: data

text: analysis

text: approaches

text: to

text: guide

text: the

text: development

text: of

text: mobile

text: interventions.

text: In

text: particular

text: data

text: from

text: mobile

text: health

text: studies

text: might

text: be

text: used

text: to

text: examine

text: eﬀect

text: moderators—

text: individual

text: characteristics,

text: time-varying

text: context

text: or

text: past

text: treatment

text: response

text: that

text: moderate

text: the

text: eﬀect

text: of

text: current

text: treatment

text: on

text: a

text: subsequent

text: response.

text: This

text: paper

text: introduces

text: a

text: formal

text: deﬁnition

text: for

text: moderated

text: eﬀects

text: in

text: terms

text: of

text: potential

text: outcomes,

text: a

text: deﬁnition

text: that

text: is

text: particularly

text: suited

text: to

text: mobile

text: interventions,

text: where

text: treatment

text: occasions

text: are

text: numerous,

text: individuals

text: are

text: not

text: always

text: available

text: for

text: treatment,

text: and

text: potential

text: moderators

text: might

text: be

text: inﬂuenced

text: by

text: past

text: treatment.

text: Methods

text: for

text: estimating

text: moderated

text: eﬀects

text: are

text: developed

text: and

text: compared.

text: The

text: proposed

text: approach

text: is

text: illustrated

text: using

text: BASICS-Mobile,

text: a

text: smartphone-based

text: intervention

text: designed

text: to

text: curb

text: heavy

text: drinking

text: and

text: smoking

text: among

text: college

text: students.

text: Keywords:

text: mHealth,

text: structural

text: nested

text: mean

text: model,

text: eﬀect

text: modiﬁcation


text: ∗ The ﬁrst simulation experiment concerns the estimation of when an important modera- β 1 tor exists. This experiment illustrates that, when primary interest is in the marginal proximal treatment eﬀect, weighting and centering is preferable over GEE. In the data generative − model, we set , , and (recall implies that the true θ = 0 η = 0.8 η = 0.8 ξ = 0 ξ = 0 2 1 2 ∗ − marginal proximal treatment eﬀect is ). Diﬀerent scenarios were devised by setting β = 0.2 1 ∗ to one of , , , giving respectively a small, medium, or large degree of moderation β 0.2 0.5 0.8 11 by . Since and are nonzero, the treatment is assigned with a probability depending S η η A t 1 2 t on both S and past treatment A − , for(cid:80)each t. t t 1 In the weighted and centered analysis, we parameterize and estimate . In particular, p˜ t − − P − p˜ (a; ρˆ) = ρˆa(1 ρˆ)1 a where ρˆ = T A /T . The weights are set to W = ρˆAt(1 t − | n t=1 t | t ρˆ)1 At/p (A H ) and the working model for E[W Y H ] is α + α S (i.e., g (H ) = t t t t t+1 t 10 11 t 1t t (cid:124)). Thus the estimating function in (6) is given by (1, S ) t (cid:32) (cid:33) (cid:88) (cid:0) (cid:1) T − − − (1, S )(cid:124) t Y (α + α S ) (A ρˆ)β W − . t+1 10 11 t t 1 t A ρˆ t=1 t A common alternative would be a GEE analysis with an independence working correlation matrix. The GEE estimating function with an independence working correlation matrix (GEE-IND) is the above estimating function but with for all and not centered. A W = 1 t A t t more likely alternate that would be used in the mobile health literature is a GEE with an non- independence working correlation matrix (Schafer 2006); the resulting conditional mean model is the same as when random eﬀects are used (Schwartz and Stone 2007; Bolger and Laurenceau 2013). We also provide a comparison with this alternative, using an AR(1) correlation matrix (GEE-AR(1)). Note that, to guarantee consistency in a GEE analysis, one would assume ∼ that the analysis model is correct; since here the analysis model is , Y α + α S + A β t+1 10 11 t t 1 | the corresponding assumption would be that for some E[Y S , A ] = α + α S + A β t+1 t t 10 11 t t 1 (α , α , β ). This assumption is false (no A S term). The weighting and(cid:2)centering method, 10 11 1 t t on the other hand, doe(cid:3)s not require a model for the conditional mean. For consistency, | the weighting and centering method only uses the assumption that E E[Y S , A = t+1 t t − | for some . 1] E[Y S , A = 0] = β β t+1 t t 1 1 Since the treatment eﬀect term does not include , the GEE conditional mean models are S t | misspeciﬁed. Furthermore since , the randomization probability depends η = 0.8 p (1 H ) 2 t t on the underlying moderator . We therefore anticipate the ˆ from the GEE methods S β t 1 15

text: The

text: ﬁrst

text: simulation

text: experiment

text: concerns

text: the

text: estimation

text: of

text: when

text: an

text: important

text: modera-

text: β

text: tor

text: exists.

text: This

text: experiment

text: illustrates

text: that,

text: when

text: primary

text: interest

text: is

text: in

text: the

text: marginal

text: proximal

text: treatment

text: eﬀect,

text: weighting

text: and

text: centering

text: is

text: preferable

text: over

text: GEE.

text: In

text: the

text: data

text: generative

text: model,

text: we

text: set

text: and

text: (recall

text: implies

text: that

text: the

text: true

text: θ

text: η

text: η

text: ξ

text: ξ

text: marginal

text: proximal

text: treatment

text: eﬀect

text: is

text: Diﬀerent

text: scenarios

text: were

text: devised

text: by

text: setting

text: β

text: to

text: one

text: of

text: giving

text: respectively

text: a

text: small,

text: medium,

text: or

text: large

text: degree

text: of

text: moderation

text: β

text: by

text: Since

text: and

text: are

text: nonzero,

text: the

text: treatment

text: is

text: assigned

text: with

text: a

text: probability

text: depending

text: S

text: η

text: η

text: A

text: t

text: t

text: on

text: both

text: S

text: and

text: past

text: treatment

text: A

text: t.

text: t

text: t

text: In

text: the

text: weighted

text: and

text: centered

text: analysis,

text: we

text: parameterize

text: and

text: estimate

text: In

text: particular,

text: p˜

text: t

text: P

text: p˜

text: (a;

text: ρˆ)

text: ρˆ)1

text: a

text: where

text: ρˆ

text: T

text: A

text: /T

text: The

text: weights

text: are

text: set

text: to

text: W

text: t

text: n

text: t

text: t

text: ρˆ)1

text: At/p

text: (A

text: H

text: and

text: the

text: working

text: model

text: for

text: E[W

text: Y

text: H

text: is

text: α

text: α

text: S

text: (i.e.,

text: g

text: (H

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: Thus

text: the

text: estimating

text: function

text: in

text: is

text: given

text: by

text: S

text: t

text: T

text: S

text: t

text: Y

text: (α

text: α

text: S

text: (A

text: ρˆ)β

text: W

text: t

text: t

text: t

text: A

text: ρˆ

text: t

text: A

text: common

text: alternative

text: would

text: be

text: a

text: GEE

text: analysis

text: with

text: an

text: independence

text: working

text: correlation

text: matrix.

text: The

text: GEE

text: estimating

text: function

text: with

text: an

text: independence

text: working

text: correlation

text: matrix

text: (GEE-IND)

text: is

text: the

text: above

text: estimating

text: function

text: but

text: with

text: for

text: all

text: and

text: not

text: centered.

text: A

text: W

text: t

text: A

text: t

text: t

text: more

text: likely

text: alternate

text: that

text: would

text: be

text: used

text: in

text: the

text: mobile

text: health

text: literature

text: is

text: a

text: GEE

text: with

text: an

text: non-

text: independence

text: working

text: correlation

text: matrix

text: (Schafer

text: the

text: resulting

text: conditional

text: mean

text: model

text: is

text: the

text: same

text: as

text: when

text: random

text: eﬀects

text: are

text: used

text: (Schwartz

text: and

text: Stone

text: Bolger

text: and

text: Laurenceau

text: We

text: also

text: provide

text: a

text: comparison

text: with

text: this

text: alternative,

text: using

text: an

text: correlation

text: matrix

text: Note

text: that,

text: to

text: guarantee

text: consistency

text: in

text: a

text: GEE

text: analysis,

text: one

text: would

text: assume

text: that

text: the

text: analysis

text: model

text: is

text: correct;

text: since

text: here

text: the

text: analysis

text: model

text: is

text: Y

text: α

text: α

text: S

text: A

text: β

text: t

text: t

text: the

text: corresponding

text: assumption

text: would

text: be

text: that

text: for

text: some

text: E[Y

text: S

text: A

text: α

text: α

text: S

text: A

text: β

text: t

text: t

text: t

text: t

text: (α

text: α

text: β

text: This

text: assumption

text: is

text: false

text: (no

text: A

text: S

text: term).

text: The

text: weighting

text: method,

text: t

text: t

text: on

text: the

text: other

text: hand,

text: not

text: require

text: a

text: model

text: for

text: the

text: conditional

text: mean.

text: For

text: consistency,

text: the

text: weighting

text: and

text: centering

text: method

text: only

text: uses

text: the

text: assumption

text: that

text: E

text: E[Y

text: S

text: A

text: t

text: t

text: for

text: some

text: E[Y

text: S

text: A

text: β

text: β

text: t

text: t

text: Since

text: the

text: treatment

text: eﬀect

text: term

text: does

text: not

text: include

text: the

text: GEE

text: conditional

text: mean

text: models

text: are

text: S

text: t

text: misspeciﬁed.

text: Furthermore

text: since

text: the

text: randomization

text: probability

text: depends

text: η

text: p

text: H

text: t

text: t

text: on

text: the

text: underlying

text: moderator

text: We

text: therefore

text: anticipate

text: the

text: ˆ

text: from

text: the

text: GEE

text: methods

text: S

text: β

text: t


text: combination of other variables. For illustrative purposes we provide an analysis acting as though the study was observational and assuming sequential ignorability; we estimate (with | logistic regression) the treatment probabilities in the denominator of the weights, , p (1 H ) t t based on urge using (Y , , 1(t < 4)) t t | − urge urge p (1 H ; ηˆ) = expit(0.69 + 0.02Y + 0.17 0.28 1(t < 4) + 0.70 1(t < 4)). t t t t t We examine proximal ( ) and lag-2 ( ) treatment eﬀects. For the proximal eﬀect k = 1 k = 2 analysis, we examine one candidate time-varying moderator incr , which indicates S = 1t t whether or not the user reported an increase in need to self-regulate thoughts over the two self-reports preceding . Thus in the estimating function (6) for the proximal eﬀect t analysis, we set incr (cid:124). For the delayed eﬀect analysis, we consider only the f (S ) = (1, ) 1t 1t t marginal lag-2(cid:80)eﬀect; thus, f(cid:80)(S ) = (1) in the estimating function (6). For both analyses, 2t 2t − − we centered and estimated the numerator of the weights based on p˜ (a; ρˆ) = ρˆa(1 ρˆ)1 a t P P where ρˆ = T I A / T I = 0.67. Hence, for both analyses, the weights were set n −t=1 −t t n |t=1 t to W = ρˆAt(1 ρˆ)1 At/p (A H ; ηˆ). In the working model for both analyses, a variety of t t t t predictors are incorporated in ( ), including an intercept term, incr , current g (H ) k = 1, 2 kt t t urge to smoke, Y − , time of day, the interaction between Y − and time of day, baseline t+1 k t+1 k smoking severity, baseline drinking level, age and gender. The data analysis leads to several conclusions. First, the mindfulness message achieved a reduction in the average next-reported smoking rate, but only when the user was experiencing − − either a stable or decreased need to self-regulate ( % CI to cigarettes per day; 95 5.45 0.15 see Table 4). Otherwise no proximal treatment eﬀect is apparent. Second, there is no evidence − to support the presence of an overall lag-2 eﬀect, with a % CI of to cigarettes 95 1.74 0.76 per day for the average reduction achieved by mindfulness treatment at the second-to-last treatment occasion. Estimated standard errors (SEs) take into account sampling error in estimated treatment probabilities (see (13) for the formula), and are corrected for small n (see ?? for details on the correction). Table 4: Proximal and lag-2 treatment eﬀects estimated from BASICS-Mobile data. Treatment eﬀect Estimate SE 95% CI p-value − − Proximal, increase in need to self-regulate 0.06 0.95 ( 1.27, 1.16) 0.99 − − − Proximal, no increase in need to self-regulate 2.80 1.29 ( 5.45, 0.15) 0.04 − − Delayed 0.49 0.61 ( 1.74, 0.76) 0.43 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: combination

text: of

text: other

text: variables.

text: For

text: illustrative

text: purposes

text: we

text: provide

text: an

text: analysis

text: acting

text: as

text: though

text: the

text: study

text: was

text: observational

text: and

text: assuming

text: sequential

text: ignorability;

text: we

text: estimate

text: (with

text: logistic

text: regression)

text: the

text: treatment

text: probabilities

text: in

text: the

text: denominator

text: of

text: the

text: weights,

text: p

text: H

text: t

text: t

text: based

text: on

text: urge

text: using

text: (Y

text: t

text: t

text: urge

text: urge

text: p

text: H

text: ηˆ)

text: t

text: t

text: t

text: t

text: t

text: We

text: examine

text: proximal

text: and

text: treatment

text: eﬀects.

text: For

text: the

text: proximal

text: eﬀect

text: k

text: k

text: analysis,

text: we

text: examine

text: one

text: candidate

text: time-varying

text: moderator

text: incr

text: which

text: indicates

text: S

text: t

text: whether

text: or

text: not

text: the

text: user

text: reported

text: an

text: increase

text: in

text: need

text: to

text: self-regulate

text: thoughts

text: over

text: the

text: two

text: self-reports

text: preceding

text: Thus

text: in

text: the

text: estimating

text: function

text: for

text: the

text: proximal

text: eﬀect

text: t

text: analysis,

text: we

text: set

text: incr

text: For

text: the

text: delayed

text: eﬀect

text: analysis,

text: we

text: consider

text: only

text: the

text: f

text: (S

text: t

text: marginal

text: thus,

text: in

text: the

text: estimating

text: function

text: For

text: both

text: analyses,

text: we

text: centered

text: and

text: estimated

text: the

text: numerator

text: of

text: the

text: weights

text: based

text: on

text: p˜

text: (a;

text: ρˆ)

text: ρˆ)1

text: a

text: t

text: P

text: P

text: where

text: ρˆ

text: T

text: I

text: A

text: T

text: I

text: Hence,

text: for

text: both

text: analyses,

text: the

text: weights

text: were

text: set

text: n

text: −t

text: t

text: n

text: t

text: to

text: W

text: ρˆ)1

text: At/p

text: (A

text: H

text: ηˆ).

text: In

text: the

text: working

text: model

text: for

text: both

text: analyses,

text: a

text: variety

text: of

text: t

text: t

text: t

text: t

text: predictors

text: are

text: incorporated

text: in

text: including

text: an

text: intercept

text: term,

text: incr

text: current

text: g

text: (H

text: k

text: kt

text: t

text: t

text: urge

text: to

text: smoke,

text: Y

text: time

text: of

text: day,

text: the

text: interaction

text: between

text: Y

text: and

text: time

text: of

text: day,

text: baseline

text: k

text: k

text: smoking

text: severity,

text: baseline

text: drinking

text: level,

text: age

text: and

text: gender.

text: The

text: data

text: analysis

text: leads

text: to

text: several

text: conclusions.

text: First,

text: the

text: mindfulness

text: message

text: achieved

text: a

text: reduction

text: in

text: the

text: average

text: next-reported

text: smoking

text: rate,

text: but

text: only

text: when

text: the

text: user

text: was

text: experiencing

text: either

text: a

text: stable

text: or

text: decreased

text: need

text: to

text: self-regulate

text: CI

text: to

text: cigarettes

text: per

text: day;

text: see

text: Table

text: Otherwise

text: no

text: proximal

text: treatment

text: eﬀect

text: is

text: apparent.

text: Second,

text: there

text: is

text: no

text: evidence

text: to

text: support

text: the

text: presence

text: of

text: an

text: overall

text: eﬀect,

text: with

text: a

text: CI

text: of

text: to

text: cigarettes

text: per

text: day

text: for

text: the

text: average

text: reduction

text: achieved

text: by

text: mindfulness

text: treatment

text: at

text: the

text: second-to-last

text: treatment

text: occasion.

text: Estimated

text: standard

text: errors

text: (SEs)

text: take

text: into

text: account

text: sampling

text: error

text: in

text: estimated

text: treatment

text: probabilities

text: (see

text: for

text: the

text: formula),

text: and

text: are

text: corrected

text: for

text: small

text: n

text: (see

text: for

text: details

text: on

text: the

text: correction).

text: Table

text: Proximal

text: and

text: treatment

text: eﬀects

text: estimated

text: from

text: BASICS-Mobile

text: data.

text: Treatment

text: eﬀect

text: Estimate

text: SE

text: CI

text: p-value

text: Proximal,

text: increase

text: in

text: need

text: to

text: self-regulate

text: Proximal,

text: no

text: increase

text: in

text: need

text: to

text: self-regulate

text: Delayed

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Klasnja, P., E. Hekler, S. Shiﬀman, A. Boruvka, D. Almirall, A. Tewari, and S. Murphy (2015). Micro-randomized trials: An experimental design for developing just-in-time adaptive interventions. Health Psychology 34 Suppl, 1220–1228. Liang, K.-Y. and S. L. Zeger (1986). Longitudinal data analysis using generalized linear models. Biometrika 73 (1), 13–22. Liao, P., P. Klasnja, A. Tewari, and S. A. Murphy (2015). Micro-randomized trials in mHealth. Statistics in Medicine. in press. Mancl, L. A. and T. A. DeRouen (2001). A covariance estimator for GEE with improved small-sample properties. Biometrics 57 (1), 126–134. Mancl, L. A. and B. G. Leroux (1996). Eﬃciency of regression estimates for clustered data. Biometrics 52 (2), 500–511. Mohr, D. C., S. M. Schueller, E. Montague, M. N. Burns, and P. Rashidi (2014). The behavioral intervention technology model: An integrated conceptual and technological framework for ehealth and mhealth interventions. Journal of Medical Internet Research 16 (6), e146. Muessig, K. E., E. C. Pike, S. Legrand, and L. B. Hightow-Weidman (2013). Mobile phone applications for the care and prevention of HIV and other sexually transmitted diseases: A review. Journal of Medical Internet Research 15 (1), e1. Neyman, J. (1990). On the application of probability theory to agricultural experiments: Essay on principles. Statistical Science 5 (4), 465–472. Translated and edited by D. M. Dabrowska and T. P. Speed. Pepe, M. S. and G. L. Anderson (1994). A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data. Communications in Statistics - Simulation and Computation 23 (4), 939–951. R Core Team (2015). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. Robins, J. M. (1989). The analysis of randomized and non-randomized AIDS treatment trials using a new approach to causal inference in longitudinal studies. In L. Sechrest, H. Freeman, A. Mulley, and National Center for Health Services Research and Health Care Technology Assessment (Eds.), Health Services Research Methodology: A Focus on AIDS, 22

text: Klasnja,

text: P.,

text: E.

text: Hekler,

text: S.

text: Shiﬀman,

text: A.

text: Boruvka,

text: D.

text: Almirall,

text: A.

text: Tewari,

text: and

text: S.

text: Murphy

text: Micro-randomized

text: trials:

text: An

text: experimental

text: design

text: for

text: developing

text: just-in-time

text: adaptive

text: interventions.

text: Health

text: Psychology

text: Suppl,

text: Liang,

text: K.-Y.

text: and

text: S.

text: L.

text: Zeger

text: Longitudinal

text: data

text: analysis

text: using

text: generalized

text: linear

text: models.

text: Biometrika

text: Liao,

text: P.,

text: P.

text: Klasnja,

text: A.

text: Tewari,

text: and

text: S.

text: A.

text: Murphy

text: Micro-randomized

text: trials

text: in

text: mHealth.

text: Statistics

text: in

text: Medicine.

text: in

text: press.

text: Mancl,

text: L.

text: A.

text: and

text: T.

text: A.

text: DeRouen

text: A

text: covariance

text: estimator

text: for

text: GEE

text: with

text: improved

text: small-sample

text: properties.

text: Biometrics

text: Mancl,

text: L.

text: A.

text: and

text: B.

text: G.

text: Leroux

text: Eﬃciency

text: of

text: regression

text: estimates

text: for

text: clustered

text: data.

text: Biometrics

text: Mohr,

text: D.

text: C.,

text: S.

text: M.

text: Schueller,

text: E.

text: Montague,

text: M.

text: N.

text: Burns,

text: and

text: P.

text: Rashidi

text: The

text: behavioral

text: intervention

text: technology

text: model:

text: An

text: integrated

text: conceptual

text: and

text: technological

text: framework

text: for

text: ehealth

text: and

text: mhealth

text: interventions.

text: Journal

text: of

text: Medical

text: Internet

text: Research

text: Muessig,

text: K.

text: E.,

text: E.

text: C.

text: Pike,

text: S.

text: Legrand,

text: and

text: L.

text: B.

text: Hightow-Weidman

text: Mobile

text: phone

text: applications

text: for

text: the

text: care

text: and

text: prevention

text: of

text: HIV

text: and

text: other

text: sexually

text: transmitted

text: diseases:

text: A

text: review.

text: Journal

text: of

text: Medical

text: Internet

text: Research

text: Neyman,

text: J.

text: On

text: the

text: application

text: of

text: probability

text: theory

text: to

text: agricultural

text: experiments:

text: Essay

text: on

text: principles.

text: Statistical

text: Science

text: Translated

text: and

text: edited

text: by

text: D.

text: M.

text: Dabrowska

text: and

text: T.

text: P.

text: Speed.

text: Pepe,

text: M.

text: S.

text: and

text: G.

text: L.

text: Anderson

text: A

text: cautionary

text: note

text: on

text: inference

text: for

text: marginal

text: regression

text: models

text: with

text: longitudinal

text: data

text: and

text: general

text: correlated

text: response

text: data.

text: Communications

text: in

text: Statistics

text: Simulation

text: and

text: Computation

text: R

text: Core

text: Team

text: R:

text: A

text: Language

text: and

text: Environment

text: for

text: Statistical

text: Computing.

text: Vienna,

text: Austria:

text: R

text: Foundation

text: for

text: Statistical

text: Computing.

text: Robins,

text: J.

text: M.

text: The

text: analysis

text: of

text: randomized

text: and

text: non-randomized

text: AIDS

text: treatment

text: trials

text: using

text: a

text: new

text: approach

text: to

text: causal

text: inference

text: in

text: longitudinal

text: studies.

text: In

text: L.

text: Sechrest,

text: H.

text: Freeman,

text: A.

text: Mulley,

text: and

text: National

text: Center

text: for

text: Health

text: Services

text: Research

text: and

text: Health

text: Care

text: Technology

text: Assessment

text: (Eds.),

text: Health

text: Services

text: Research

text: Methodology:

text: A

text: Focus

text: on

text: AIDS,


text: Tchetgen Tchetgen, E. J., M. M. Glymour, J. Weuve, and J. Robins (2012). Specifying the correlation structure in inverse-probability-weighting estimation for repeated measures. Epidemiology 23 (4), 644–646. Letter to the Editor. Vanderweele, T. J., G. Hong, S. M. Jones, and J. L. Brown (2013). Mediation and spillover eﬀects in group-randomized trials: A case study of the 4rs educational intervention. Journal of the American Statistical Association 108 (502), 469–482. Vansteelandt, S. (2007). On confounding, prediction and eﬃciency in the analysis of lon- gitudinal and cross-sectional clustered data. Scandinavian Journal of Statistics 34 (3), 478–498. Wang, L., A. Rotnitzky, X. Lin, R. E. Millikan, and P. F. Thall (2012). Evaluation of viable dynamic treatment regimes in a sequentially randomized trial of advanced prostate cancer. Journal of the American Statistical Association 107 (498), 493–508. Witkiewitz, K., S. A. Desai, S. Bowen, B. C. Leigh, M. Kirouac, and M. E. Larimer (2014). Development and evaluation of a mobile intervention for heavy drinking and smoking among college students. Psychology of Addictive Behaviors 28 (3), 639–650. Acknowledgements: Funding was provided by the National Institute on Drug Abuse (P50DA039838, R01DA039901, R01DA015697), National Institute on Alcohol Abuse and Alcoholism (R01AA023187), National Heart Lung and Blood Institute (R01HL125440), and National Institute of Biomedi- cal Imaging and Bioengineering (U54EB020404). 24

text: Tchetgen

text: Tchetgen,

text: E.

text: J.,

text: M.

text: M.

text: Glymour,

text: J.

text: Weuve,

text: and

text: J.

text: Robins

text: Specifying

text: the

text: correlation

text: structure

text: in

text: inverse-probability-weighting

text: estimation

text: for

text: repeated

text: measures.

text: Epidemiology

text: Letter

text: to

text: the

text: Editor.

text: Vanderweele,

text: T.

text: J.,

text: G.

text: Hong,

text: S.

text: M.

text: Jones,

text: and

text: J.

text: L.

text: Brown

text: Mediation

text: and

text: spillover

text: eﬀects

text: in

text: group-randomized

text: trials:

text: A

text: case

text: study

text: of

text: the

text: educational

text: intervention.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association

text: Vansteelandt,

text: S.

text: On

text: confounding,

text: prediction

text: and

text: eﬃciency

text: in

text: the

text: analysis

text: of

text: lon-

text: gitudinal

text: and

text: cross-sectional

text: clustered

text: data.

text: Scandinavian

text: Journal

text: of

text: Statistics

text: Wang,

text: L.,

text: A.

text: Rotnitzky,

text: X.

text: Lin,

text: R.

text: E.

text: Millikan,

text: and

text: P.

text: F.

text: Thall

text: Evaluation

text: of

text: viable

text: dynamic

text: treatment

text: regimes

text: in

text: a

text: sequentially

text: randomized

text: trial

text: of

text: advanced

text: prostate

text: cancer.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association

text: Witkiewitz,

text: K.,

text: S.

text: A.

text: Desai,

text: S.

text: Bowen,

text: B.

text: C.

text: Leigh,

text: M.

text: Kirouac,

text: and

text: M.

text: E.

text: Larimer

text: Development

text: and

text: evaluation

text: of

text: a

text: mobile

text: intervention

text: for

text: heavy

text: drinking

text: and

text: smoking

text: among

text: college

text: students.

text: Psychology

text: of

text: Addictive

text: Behaviors

text: Acknowledgements:

text: Funding

text: was

text: provided

text: by

text: the

text: National

text: Institute

text: on

text: Drug

text: Abuse

text: National

text: Institute

text: on

text: Alcohol

text: Abuse

text: and

text: Alcoholism

text: National

text: Heart

text: Lung

text: and

text: Blood

text: Institute

text: and

text: National

text: Institute

text: of

text: Biomedi-

text: cal

text: Imaging

text: and

text: Bioengineering