Directed Acyclic Graphs (DAGs) for Causal Inference

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DAGs & EMM

Prep

:

notes | HW1-4 | scratchpad

Install packages:

(1) dagitty, (2) ggdag, (3) contrast

2020.02.27 – L15 – Mike

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•

DAGs

•

EMM

•

More on models

•

Homework 3

: Looked good!

•

Homework 4

: Basic GLMs

DAGs

Models

head

(

births_sm

)

births_sm

$

mage_sq 

=

 births_sm

$

mage 

^

2 # or poly

m1 

=

glm

(

formula

=

 preterm_f 

~

 pnc5_f,

family

=

binomial

(

"identity"

)

, 

data

=

 births_sm

)

m2 

=

glm

(

formula

=

 preterm_f 

~

 pnc5_f 

+

smoker_f 

,

family

=

binomial

(

"identity"

)

, 

data

=

 births_sm

)

m3 

=

glm

(

formula

=

 preterm_f 

~

 pnc5_f 

+

 smoker_f 

+

mage

,

family

=

binomial

(

"identity"

)

, 

data

=

 births_sm

)

m4 

=

glm

(

formula

=

 preterm_f 

~

 pnc5_f 

+

 smoker_f 

+

poly

(

mage, 

2

, 

raw

=

 T

)

, 

family

=

binomial

(

"identity"

)

, 

data

=

 births_sm

)

Our (toy) DAG

Directed Acyclic Graphs (DAGs)

 inform our variable selection and treatment in

models (based on their status as mediators, confounders, effect measure

modifiers, etc. We will not elaborate in this class!

Take the Epi sequence for more.

DAG from EPID 716 / Christy Avery

What is a DAG?

DAGs

 (

Directed Acyclic Graphs

) document causal

assumptions / knowledge from our head or

literature. They can be used to guide us to better

answer questions like:

•

Does a change in A prompt a change in B?

•

Or the other way around?

•

Or not in either direction, but their association is

caused by some third thing?

DAG Requirements

Directed

: 



, not --

Acyclic

: A



B, not A



B and B



A

Graph

: Connected, not dangling*.

Other disciplines, from engineering to other forms of

statistical models, relax these requirements.

Mediation

Confounders

Colliders

Inducing a biased causal association through

controlling a collider is: 

collider stratification bias

Effect Measure Modifiers

EMM: Two notes

•

Note that an 

EMM may or may not be a confounder

(influencing the values of A and B directly, vs. the effect

of A on B), so may not be in the DAG node network.

EMMs are good for a DAG notes though!

•

Also note that it may be rare that an exposure /

intervention switches direction entirely. Effect measure

modification worth acknowledging may be a matter of

degree

… or important to report because of 

context -

regardless of statistical interaction, p=whatever.

In Sum:

Create a model, throw things in, reduce by p-value /

backwards selection, or any of a number of techniques.

May be good at predicting outcome from exposure and

other variables. There is nuance here, but…

In Sum:

…if we want the causal effect, we have to be intentional

about what parts of this flow we block. We leave direct

and indirect causal paths, and leave blocked paths with

colliders (do not control!)

How do we do this?

•

Encode the nodes and directed edges of the DAG

from the literature / content knowledge

•

By eye, hand, or software

1.

document all paths between Exposure and Outcome

2.

Identify whether they are already blocked (collider),

backdoor (confounded), or causal (direct or indirect

through a mediator)

3.

Select nodes to statistically control, often ideally as

few as possible, to block the open backdoor paths

without blocking

A note on functional form!

Remember 

maternal age

? In order to improve precision

of estimates (and acknowledge the linear assumptions of

GLMs), it behooves us to model covariates as well as

possible… balancing parsimony and interpretation.

Hence mage2… or splines of some kind.

This does not apply as directly to our exposures, which

we want to interpret in actionable, communicable ways!

….and a note on interpretation!

Relatedly, the Table 2 fallacy suggests

Reminder: Table 2 fallacy! (Westrich & Greenland 2013).

If those “estimates” are largely uninterpretable in causal

inference context, might as well let them go and model

them more precisely (albeit obfuscated).

Westreich, D., and S. Greenland. “The Table 2 Fallacy: Presenting and Interpreting Confounder and

Modifier Coefficients.” 

American Journal of Epidemiology

 177, no. 4 (February 15, 2013): 292–98.

https://doi.org/10.1093/aje/kws412

.

DAG critiques

•

Reality isn’t DAGGY

•

DAGs (even if large) are a model, and so a simplified version of reality,

in this particular case requiring unidirectionality and acyclic

assumptions.

•

Reality is often a system with feedback loops and inter-relationships

that may not be modeled well with unidirectional models, perhaps

especially with social / network processes. There are other methods!

•

Not all nodes / edges are alike

•

Race-ethnicity in particular is a heavily overloaded construct for causal

inference (Vanderweele & Robinson 2014), reaching back to represent

historical and current systemic oppression and racism, physical

phenotype, experiences of cultural and ascribed identity. Parts of this

construct may have different causal relationships. Break apart if you

can, and regardless, be mindful / nuanced in your interpretation.

•

Not always interested in causal effects

•

Prediction, classification, association, other techniques have a place in

a public health toolbox.

VanderWeele, Tyler J., and Whitney R. Robinson. “On the Causal Interpretation of Race in Regressions Adjusting for Confounding and

Mediating Variables:” 

Epidemiology

 25, no. 4 (July 2014): 473–84. doi:10.1097/EDE.0000000000000105.

Our (toy) DAG

Directed Acyclic Graphs (DAGs)

 inform our variable selection and treatment in

models (based on their status as mediators, confounders, effect measure

modifiers, etc. We will not elaborate in this class!

Take the Epi sequence for more.

DAG from EPID 716 / Christy Avery

Let’s Try: Dagitty

•

Check it out here: 

www.dagitty.net

•

Premade DAG for you: 

dagitty.net/moAh6a6

DAGs in R

The online 

dagitty 

tool

(

http://www.dagitty.net/dags.html#

) exports R code

for the dagitty package. Can be directly downloaded

into R!

The new 

ggdag 

package is a beefed-up version using

tidy data structures we can recognize.

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e.g.: 

http://www.dagitty.net/dags.html?id=oAh6a6

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library

(

dagitty

)

birth_daggity 

=

 downloadGraph

(

"dagitty.net/moAh6a6"

)

# ^ From the internet!

birth_daggity

plot

(

birth_daggity

)

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# Imagine! run a model driven by a DAG! Or tweak the dag and

rerun the model!  :p

>

 dagitty

::

adjustmentSets

(

birth_daggity

)

{

 Maternal_Age, Maternal_Smoking, Race_Ethnicity 

}

>

 minimal_adjustment_set 

=

dagitty

::

adjustmentSets

(

birth_daggity, type

=

"minimal"

)[[

1

]]

>

 minimal_adjustment_set

[

1

]

"Maternal_Age"

"Maternal_Smoking"

"Race_Ethnicity"

>

str

(

birth_daggity

)

# < DAGs are messy though. :/

'dagitty'

 Named chr 

"dag {\nChild_Gender [pos=\"0.111,-

0.146\"]\nEarlyPNC [exposure,pos=\"-0.366,-

0.090\"]\nMaternal_Age [pos=\"-0.5"

|

 __truncated__

>

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 tidy_dagitty()

•

 ggdag(dag_df) + theme_dag() gets you a nice

ggplot2

•

 geometries, edit plot as usual

•

See: 

https://cran.r-

project.org/web/packages/ggdag/vignettes/intro-

to-ggdag.html

•

And: 

https://github.com/malcolmbarrett/ggdag

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>

library

(

ggdag

)

>

>

 dag_df 

=

 tidy_dagitty

(

birth_daggity

)

>

str

(

dag_df

)

List of 

2

$

data

:

Classes ‘tbl_df’, ‘tbl’ and 

'data.frame'

:

15

 obs. of  

8

 variables

:

  ..

$

 name     

:

 chr 

[

1

:

15

]

"Child_Gender"

"EarlyPNC"

"Maternal_Age"

"Maternal_Age"

 ...

  ..

$

 x        

:

 num 

[

1

:

15

]

0.111

-

0.366

-

0.576

-

0.576

-

0.576

-

0.48

-

0.48

-

0.619

-

0.619

-

0.619

 ...

  ..

$

 y        

:

 num 

[

1

:

15

]

-

0.146

-

0.09

-

0.091

-

0.091

-

0.091

0.022

0.022

-

0.051

-

0.051

-

0.051

 ...

  ..

$

 direction

:

 Factor w

/

3

levels

"<-"

,

"->"

,

"<->"

:

2

2

2

2

2

2

2

2

2

2

 ...

  ..

$

 to       

:

 chr 

[

1

:

15

]

"Preterm_Birth"

"Preterm_Birth"

"EarlyPNC"

"Maternal_Smoking"

 ...

  ..

$

 xend     

:

 num 

[

1

:

15

]

-

0.147

-

0.147

-

0.366

-

0.48

-

0.147

-

0.366

-

0.147

-

0.366

-

0.576

-

0.48

 ...

  ..

$

 yend     

:

 num 

[

1

:

15

]

-

0.051

-

0.051

-

0.09

0.022

-

0.051

-

0.09

-

0.051

-

0.09

-

0.091

0.022

 ...

  ..

$

 circular 

:

 logi 

[

1

:

15

]

FALSE

FALSE

FALSE

FALSE

FALSE

FALSE

 ...

$

 dag 

:

'dagitty'

 Named chr 

"dag {\nChild_Gender [pos=\"0.111,-0.146\"]\nEarlyPNC

[exposure,pos=\"-0.366,-0.090\"]\nMaternal_Age [pos=\"-0.5"

|

 __truncated__

-

attr

(*

, 

"class"

)=

 chr 

"tidy_dagitty"

>

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plot

(

dag_df

)

# nope!

ggdag

(

dag_df 

%>% mutate(name = str_trunc(name, 5))

)

+

  theme_dag_blank

()

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ggdag_adjustment_set

(

dag_df

)

+

  theme_dag_blank

()

+

# Hey that's nice!

  geom_dag_text

(

color

=

"black"

)

# gives us geometries as usual

Practical Uses

R may be helpful for quickly changing and rerunning

for minimal sets a few different DAG scenarios.

Probably better than hand. Nice pair with dagitty

website.

Maybe useful for SEM? I dunno. But you can stick

your DAGs in papers / R Markdown, make ggplots of

them, etc.

Other Packages

If you do this a lot, find your favorite! R skills let you translate

data structures across packages as you need to.

dagitty  * ggdag

: Today!

DiagrammeR

: Prettier, but no adjustment sets?

https://donlelek.github.io/2015-03-31-dags-with-r/

dagR

: Prettier and adjustment sets, but funny syntax?

http://rstudio-pubs-

static.s3.amazonaws.com/2609_e3d86d0748c04eb18d5f56d6

a99feb3f.html

Maybe new packages?

EMM

Effect Measure Modifiers

Crude, but like…

Another view…

In other words

A continuation of modeling:

How do we get 

group-specific effect

estimates

 and 

confidence intervals

from models when we have EMM?

Also: 

a recursion primer

Review

1.

We’ve looked at a 

Directed Acyclic Graph

 to identify

model covariates to control to assess the causal

relationship of early prenatal care on preterm birth.

We know EMM won’t show up on the DAG.

2.

We’ve constructed 

Generalized Linear Models 

for

risk difference (also looked at RRs and ORs) without

EMM.

3.

We’ve looked at 

crude EMM 

using dplyr:: and table().

Moderate modification?

4.

Now we want modeled, stratum-specific effect

estimates and CIs controlling for our other covariates

(smoking, maternal age). We want a 

full EMM

model

.

Model Specification 1: Crude

Model Specification 2

Model Specification 3

Model Specification 3

Full Model Specification w/ EMM

Coding schemes

R’s default for expanding unordered factors in a

model is 

indicator coding

:

R breaks up factors in models to all be referent to a

baseline level (set by factor() or fct_* functions!).

This means you need 

(n-1) indicator levels 

to

describe 

n levels of a factor

.

Let’s Look: Contrast Matrix

Full Model Specification w/ EMM

Full Model Specification w/ EMM

map(births[, map_lgl(births, is.factor)], levels)

# ^ level check – purrr

mfull_emm_rd 

=

glm

(

data

=

births,

                   preterm_f 

~

pnc5_f 

*

 raceeth_f 

+

                     smoker_f 

+

 mage 

+

 mage_sq,

family

=

binomial

(

link

=

"identity"

))

Detail

>

 mfull_emm_rd 

=

glm

(

data

=

births,

+

                    preterm_f 

~

 pnc5_f 

*

 raceeth_f 

+

+

                      smoker_f 

+

 mage 

+

 mage_sq,

+

family

=

binomial

(

link

=

"identity"

))

>

 broom

::

tidy

(

mfull_emm_rd

)

# A tibble: 13 x 5

   term                                                       estimate std.error statistic   p.value

<

chr

>

<

dbl

>

<

dbl

>

<

dbl

>

<

dbl

>

1

(

Intercept

)

0.725

0.0249

29.1

9.21

e

-

187

2

 pnc5_fEarly PNC                                            

0.0221

0.00683

3.23

1.22

e

-

3

3

 raceeth_fOther                                            

-

0.0130

0.0112

-

1.17

2.44

e

-

1

4

 raceeth_fWhite Hispanic                                   

-

0.0104

0.0246

-

0.422

6.73

e

-

1

5

 raceeth_fAfrican American                                 

-

0.0589

0.0109

-

5.41

6.15

e

-

8

6

 raceeth_fAmerican Indian or Alaska Native                 

-

0.0673

0.0416

-

1.62

1.06

e

-

1

7

 smoker_fSmoker                                            

-

0.0331

0.00427

-

7.77

7.96

e

-

15

8

 mage                                                       

0.0135

0.00174

7.78

7.15

e

-

15

9

 mage_sq                                                   

-

0.000246

0.0000306

-

8.02

1.09

e

-

15

10

 pnc5_fEarly PNC

:

raceeth_fOther                            

-

0.00570

0.0117

-

0.489

6.25

e

-

1

11

 pnc5_fEarly PNC

:

raceeth_fWhite Hispanic                    

0.0113

0.0256

0.440

6.60

e

-

1

12

 pnc5_fEarly PNC

:

raceeth_fAfrican American                  

0.00353

0.0114

0.310

7.56

e

-

1

13

 pnc5_fEarly PNC

:

raceeth_fAmerican Indian or Alaska Native  

0.0213

0.0433

0.492

6.23

e

-

1

>

table

(

interaction

(

births

$

pnc5_f, births

$

raceeth_f

))

# : = interaction()

              No Early PNC.White non

-

Hispanic                  Early PNC.White non

-

Hispanic

2105

32523

                           No Early PNC.Other                               Early PNC.Other

1240

8626

                  No Early PNC.White Hispanic                      Early PNC.White Hispanic

175

1387

                No Early PNC.African American                    Early PNC.African American

1836

12768

No Early PNC.American Indian or Alaska Native    Early PNC.American Indian or Alaska Native

86

783

Full Model Specification w/ EMM

Contrasts

We can get these point-estimates

by hand a few ways…

# Point estimates by hand, a few ways

mfull_emm_rd

$

coefficients

[

2

]

#WnH

RD_WnH 

=

 mfull_emm_rd

$

coefficients

[

"pnc5_fPNC starts in first 5 mo"

]

#WnH

RD_WnH 

+

 mfull_emm_rd

$

coefficients

[

"pnc5_fPNC starts in first 5 mo:raceeth_fAA"

]

#AA

sum

(

mfull_emm_rd

$

coefficients

[

c

(

2

,

10

)])

# AA

sum

(

mfull_emm_rd

$

coefficients

[

c

(

2

,

11

)])

# WH

sum

(

coef

(

mfull_emm_rd

)[

c

(

2

,

12

)])

# AI/AN

sum

(

coef

(

mfull_emm_rd

)[

c

(

2

,

13

)])

# Other

…But what about the CIs?

Can't quite make it from the individual coefficients.

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Lets you define arbitrary contrasts across your model

that you’re interested in by setting two coefficient

specifications as lists.

contrast(<MODEL OBJECT>,

         a=list(<COEFA>), b=list(<COEFB>)

Handles more than GLMs – also multi-level/LME, gee,

etc. If you need exponentiated estimates, pass in

fun=exp. See help page

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For our full model…

Gets us the RD effect estimate and 95% CI for AA moms getting prenatal care vs.

those who aren’t.

Note that unchanging model levels are required to have something, but “drop out”

of the estimation. 

This is similar to how predict() works.

# Contrasts using contrast package

library

(

contrast

)

contrast

(

mfull_emm_rd,

         a

=

list

(

pnc5_f 

=

"PNC starts in first 5 mo"

,

raceeth_f 

=

"AA"

, smoker_f 

=

"Smoker"

,

mage

=

20

, mage_sq

=

400

)

,

         b

=

list

(

pnc5_f 

=

"No PNC before 5 mo"

,

raceeth_f 

=

"AA"

, smoker_f

=

"Smoker"

,

mage

=

20

, mage_sq

=

400

))

C

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E

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For all levels of EMM:

Gets us the RD effect estimate and 95% CI for all levels of race-ethnicity including

referent. 

Can use data.frame() to piece together a results table for graphing.

raceeth_diff 

=

 contrast

(

mfull_emm_rd,

         a

=

list

(

pnc5_f 

=

"PNC starts in first 5 mo"

,

raceeth_f 

=

levels

(

births

$

raceeth_f

)

,

smoker_f 

=

"Smoker"

, mage

=

20

, mage_sq

=

400

)

,

         b

=

list

(

pnc5_f 

=

"No PNC before 5 mo"

,

raceeth_f 

=

levels

(

births

$

raceeth_f

)

,

smoker_f

=

"Smoker"

, mage

=

20

, mage_sq

=

400

))

print

(

raceeth_diff, X

=

T

)

EMM_df 

=

 data.frame

(

model

=

paste0

(

"M5: "

, raceeth_diff

$

raceeth_f

)

,

                    estimate

=

raceeth_diff

$

Contrast, std.error 

=

 raceeth_diff

$

SE,

                   X2.5..

=

raceeth_diff

$

Lower, X97.5..

=

raceeth_diff

$

Upper

)

ggplot

(

bind_rows

(

model_results,EMM_df

)

,

aes

(

model, estimate,color

=

std.error, fill

=

std.error

))+

  geom_linerange

(

aes

(

ymin

=

X2.5.., ymax

=

X97.5..

)

, size

=

1

)+

  geom_point

(

shape

=

15

, size

=

4

, color

=

"white"

)+

 geom_point

(

shape

=

15

)+

  scale_y_continuous

(

limits

=

c

(

NA

,

NA

)

, breaks

=

seq

(-

0.15

, 

0.05

, 

0.01

))+

  geom_hline

(

yintercept 

=

0

, lty

=

2

, color

=

"grey"

)+

  geom_vline

(

xintercept 

=

4.5

, lty

=

2

, color

=

"blue"

)+

annotate

(

geom 

=

"text"

, x 

=

4.6

, y

=-

0.05

, angle

=

90

,

label

=

"M5: Race-Ethnicity EMM: Stratum-Specific RDs"

)+

  labs

(

title

=

"Model results"

, subtitle

=

"Mirroring tufte boxplot aesthetics,

see ggthemes::geom_tufteboxplot()"

)+

  ggthemes

::

theme_tufte

()

#............................................

# EMM / final model

#............................................

glm

(

data

=

births, preterm_f 

~

 pnc5_f 

+

 smoker_f, 

family

=

binomial

(

link

=

"identity"

))

#Discussion of model spec

head

(

births

)

table

(

births

$

race_f, births

$

methnic

)

# note my "ethnicity" coding may not be ideal.

mfull_emm_rd 

=

glm

(

data

=

births,

                   preterm_f 

~

 pnc5_f 

*

 raceeth_f 

+

                     smoker_f 

+

 mage 

+

 mage_sq,

family

=

binomial

(

link

=

"identity"

))

summary

(

mfull_emm_rd

)

broom

::

tidy

(

mfull_emm_rd

)

# Point estimates by hand, a few ways

mfull_emm_rd

$

coefficients

[

2

]

#WnH

RD_WnH 

=

 mfull_emm_rd

$

coefficients

[

"pnc5_fPNC starts in first 5 mo"

]

#WnH

RD_WnH 

+

 mfull_emm_rd

$

coefficients

[

"pnc5_fPNC starts in first 5 mo:raceeth_fAA"

]

#AA

sum

(

mfull_emm_rd

$

coefficients

[

c

(

2

,

10

)])

# AA

sum

(

mfull_emm_rd

$

coefficients

[

c

(

2

,

11

)])

# WH

sum

(

coef

(

mfull_emm_rd

)[

c

(

2

,

12

)])

# AI/AN

sum

(

coef

(

mfull_emm_rd

)[

c

(

2

,

13

)])

# Other

C

(

births

$

raceeth_f

)

# get or set contrasts

contrasts

(

births

$

raceeth_f

)

# Let's look: contrast matrix

# See contr.treatment or contr.sum

# Contrasts using contrast package

library

(

contrast

)

contrast

(

mfull_emm_rd,

         a

=

list

(

pnc5_f 

=

"PNC starts in first 5 mo"

, raceeth_f 

=

"AA"

, smoker_f 

=

"Smoker"

, mage

=

20

, mage_sq

=

400

)

,

         b

=

list

(

pnc5_f 

=

"No PNC before 5 mo"

, raceeth_f 

=

"AA"

, smoker_f

=

"Smoker"

, mage

=

20

, mage_sq

=

400

))

raceeth_diff 

=

 contrast

(

mfull_emm_rd,

         a

=

list

(

pnc5_f 

=

"PNC starts in first 5 mo"

, raceeth_f 

=

levels

(

births

$

raceeth_f

)

, smoker_f 

=

"Smoker"

, mage

=

20

, mage_sq

=

400

)

,

         b

=

list

(

pnc5_f 

=

"No PNC before 5 mo"

, raceeth_f 

=

levels

(

births

$

raceeth_f

)

, smoker_f

=

"Smoker"

, mage

=

20

, mage_sq

=

400

))

print

(

raceeth_diff, X

=

T

)

EMM_df 

=

 data.frame

(

model

=

paste0

(

"M5: "

, raceeth_diff

$

raceeth_f

)

,

                    estimate

=

raceeth_diff

$

Contrast, std.error 

=

 raceeth_diff

$

SE,

                   X2.5..

=

raceeth_diff

$

Lower, X97.5..

=

raceeth_diff

$

Upper

)

ggplot

(

bind_rows

(

model_results,EMM_df

)

, aes

(

model, estimate,color

=

std.error, fill

=

std.error

))+

  geom_linerange

(

aes

(

ymin

=

X2.5.., ymax

=

X97.5..

)

, size

=

1

)+

  geom_point

(

shape

=

15

, size

=

4

, color

=

"white"

)+

 geom_point

(

shape

=

15

)+

  scale_y_continuous

(

limits

=

c

(

NA

,

NA

)

, breaks

=

seq

(-

0.15

, 

0.05

, 

0.01

))+

  geom_hline

(

yintercept 

=

0

, lty

=

2

, color

=

"grey"

)+

  geom_vline

(

xintercept 

=

4.5

, lty

=

2

, color

=

"blue"

)+

  annotate

(

geom 

=

"text"

, x 

=

4.6

, y

=-

0.05

, angle

=

90

, label

=

"M5: Race-Ethnicity EMM: Stratum-Specific RDs"

)+

  labs

(

title

=

"Model results"

, subtitle

=

"Mirroring tufte boxplot aesthetics, see ggthemes::geom_tufteboxplot()"

)+

  ggthemes

::

theme_tufte

()

Code

chunk

Other ways

Specifying the contrast type using functions like:

C() 

: Sets the "contrasts" attribute for the factor.

 contrasts(): 

Set and view the contrasts associated with a factor.

 model.matrix() 

: creates a design (or model) matrix, e.g., by

expanding factors to a set of dummy variables (depending on the

contrasts) and expanding interactions similarly.

Create a custom contrast matrix so that individual

coefficients are more meaningful under EMM. Possible

but beyond scope of this class. Good luck. 



References

A (sort of) complete guide to contrasts in R

http://rstudio-pubs-static.s3.amazonaws.com/65059_586f394d8eb84f84b1baaf56ffb6b47f.html

contrast 

package

https://cran.r-project.org/web/packages/contrast/vignettes/contrast.pdf

D

o

n

e

!

Coming up next week

Outputs / Reports /

Markdown

Slide

Template

Bank

P

o

s

t

-

C

h

e

c

k

Y

o

u

t

r

y

!

O

p

e

n

R

a

n

d

…

1)

Thing 1

2)

Thing 2

3)

Head to the google doc and

type “Done” next to your

name when done!

P

r

e

-

C

h

e

c

k

/

P

o

s

t

-

C

h

e

c

k

/

P

a

u

s

e

&

C

h

e

c

k

A

n

s

w

e

r

i

n

t

h

e

g

o

o

g

l

e

d

o

c

…

After 

doing a thing

, enter

something

A.

B.

C.

Example: Type order (e.g. BCA)

into google doc.

Small Group Activity

Template

•

A

•

B

•

C