Power4CluteredData

Resources used to create the lab:

This document is created mimicking couple of tutorials 1. Ben Whaley’s Tutorial https://benwhalley.github.io/just-enough-r/power-analysis.html 2. Hamburg’s Tutorial that can be found at the below link https://humburg.github.io/Power-Analysis/simr_power_analysis.html 3. Simr package vignette https://cran.r-project.org/web/packages/simr/vignettes/examples.html

Example study:

The first step in creating a model for simulation is to create a set of covariates that the model will be based on.

Data on subjects/patients from different hospitals/centres in North America will be collected. For each participant measurements will be taken on the following:

1. Palliative Care Outcomes Length of ICU stay ( continuous outcome), Whether Died within 180 days (binary outcome), etc.

2. Educational Intervention Health care proxy, Advanced Directive, fraility assesment etc.,

We start by setting up a data frame with 25 centres and 40-100 subjects per centre.

For now we will use an inbuilt simulated data, before we can create a dataset as described above.

Fitting a model using simulated data a.k.a. simdata from simr package

simdata data set,which is included in the package, mimicks environmental monitoring data, with a response variable z (e.g. bird abundance) measured at 10 levels of the continuous fixed effect variable x (e.g. study year) for three groups g (e.g. study site). There is also a continuous response variable y.

We start by inspecting the data

#head(simdata)
#dt <- simdata[1:6, 1:4]
#kbl(dt)
#html scrollable version
kbl(cbind(simdata)) %>%
  kable_paper() %>%
  scroll_box(width = "500px", height = "200px")

y	x	g	z
8.139081	1	a	3
7.947861	2	a	3
9.283638	3	a	3
7.779489	4	a	2
5.803512	5	a	3
6.131720	6	a	2
6.122911	7	a	3
7.542655	8	a	0
6.428387	9	a	2
6.219345	10	a	1
9.170150	1	b	2
8.253626	2	b	3
10.346381	3	b	2
8.807318	4	b	1
6.092850	5	b	1
8.510823	6	b	3
7.086676	7	b	2
6.241644	8	b	3
6.841493	9	b	0
5.783463	10	b	2
13.697234	1	c	9
13.551086	2	c	7
12.239432	3	c	6
14.513912	4	c	2
13.579498	5	c	2
12.037988	6	c	4
14.179940	7	c	3
13.102589	8	c	4
12.131054	9	c	2
13.071251	10	c	2

Notes on choice of multilevel model

To use simr:: we first run our ‘model of interest’. In this case it’s a random-intercepts multilevel model. We can verify x is unrelated to outcome with lmerTest::anova:

When data are clustered then multilevel/ linear mixed models, serve two purposes:

1. They overcome limitations of conventional models which assume that data are independently sampled

2. They allow us to estimate how much variation in the outcome is due to different levels of the nested structure. For example one might want to know how much influence hospital/trasuma centre have on their subjects/patients outcome. Therefore, if patients are ‘nested’ within hospitals then multilevel models can estimate the variation between hospitals (the ‘hospital effect’) and variation ‘within’ hospital ( i.e. variation between patients).

For further details please check out 1. RShowDoc(“lmerperf”, package = “lme4”) for [examples] 2. https://cran.r-project.org/web/packages/lme4/vignettes/lmer.pdf

Model Specs x + (x | g) or 1 + x + (1 + x | g) Correlated random intercept and slope.

We start by fitting a very simple Poisson mixed effects model in lme4 to the simdata data set. In this case we have a random intercept model, where each group (g) has its own intercept but the groups share a common trend.

model1 <- glmer(z ~ x + (1|g), family="poisson", data=simdata)

summary(model1)

## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula: z ~ x + (1 | g)
##    Data: simdata
## 
##      AIC      BIC   logLik deviance df.resid 
##    109.0    113.2    -51.5    103.0       27 
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.28918 -0.41836 -0.03916  0.57284  1.29631 
## 
## Random effects:
##  Groups Name        Variance Std.Dev.
##  g      (Intercept) 0.08345  0.2889  
## Number of obs: 30, groups:  g, 3
## 
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  1.54079    0.27173   5.670 1.43e-08 ***
## x           -0.11481    0.03955  -2.903   0.0037 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##   (Intr)
## x -0.666

anova(model1)

## Analysis of Variance Table
##   npar Sum Sq Mean Sq F value
## x    1 8.3215  8.3215  8.3215

extract_eq(model1)

\[ \begin{aligned} \operatorname{z}_{i} &\sim \operatorname{Poisson}(\lambda_i) \\ \log(\lambda_i) &=\alpha_{j[i]} + \beta_{1}(\operatorname{x}) \\ \alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right) \text{, for g j = 1,} \dots \text{,J} \end{aligned} \]

Specifying an Effect Size

We tweak the fitted parameters to represent our predicted values for effect sizes, variances, and covariances.

The estimated effect size for x (trend in x) is −0·11, which is significant at the 0·01 level using the default z-test. Note this is just beginier level based on the linked tutorial

Later we will incorporatea larger number of groups, and would consider problems such as overdispersion.

Before starting a power analysis, it is important to consider what sort of effect size one is interested in.

1. Power generally increases with effect size, with larger effects being easier to detect.
2. Retrospective ‘observed power’ calculations, where the target effect size comes from the data, give misleading results (Hoenig & Heisey 2001).

We will now check the effect size which can be interpreted according to Chen, Cohen, and Chen (2010)[https://slcladal.github.io/pwr.html#ref-chen2010big]

To inspect the effect sizes in the model, we determine the odds ratios of the fixed effects and then convert them into Cohen’s d values for which we have associations between traditional denominations (small, medium, and large) and effect size values.

1. small effect (Cohen’s d 0.2, OR = 1.68)

2. medium effect (Cohen’s d 0.5, OR = 3.47)

3. strong effect (Cohen’s d 0.8, OR = 6.71)

# extract fixed effect estimates
estimatesfixedeffects <- fixef(model1)
estimatesfixedeffects

## (Intercept)           x 
##   1.5407931  -0.1148147

# convert estimates into odds ratios
exp(estimatesfixedeffects)

## (Intercept)           x 
##   4.6682913   0.8915313

For example to find the power to detect a slope of −0·05. The size of the fixed effect for the variable x can be changed from −0·11 to −0·05 as follows:

Note We only change the fixed slope for the variable x. However, we could also change the random effect parameters or the residual variance. See the help entry ?modify for more details.

fixef(model1)["x"]

##          x 
## -0.1148147

## x 

## -0.1148147

fixef(model1)["x"] <- -0.05

Power Analysis via simulation

We now use the powerSim function to use our tweaked model to:

1 Create a dataset using the parameters of our model (i.e. make random draws of y which relate to g and x as specified in the model summary).

2. Re-run lmer on this simulated data.

3. Repeat this hundreds or thousands of times

4. Count how many times (i.e., for what proportion) we get a significant p value

Our observed power (proportion of times we get a significant p value) is very low here, so we might want increase our hypothesised effect of x, for example to see what power we have to detect an effect of x = -.05 as above.

set.seed(123)

powerSim(model1,nsim=20)

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## Power for predictor 'x', (95% confidence interval):
##       25.00% ( 8.66, 49.10)
## 
## Test: z-test
##       Effect size for x is -0.050
## 
## Based on 20 simulations, (0 warnings, 0 errors)
## alpha = 0.05, nrow = 30
## 
## Time elapsed: 0 h 0 m 0 s

Note If sample size is small dont use z-test ,use parametric bootstrap test instead. For examples of different test specifications, use help entry ?tests.

Increasing the sample size

In the above example, estimated power is very low ( about 32%). In simr, the extend function can be used to add rows to a data frame.

The pilot study had observations at 10 values of x, representing for example study years 1 through 10. In this step, we will calculate the effect of increasing this to 50 years/later subjects.

## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula: z ~ x + (1 | g)
##    Data: simdata
## 
##      AIC      BIC   logLik deviance df.resid 
##    109.0    113.2    -51.5    103.0       27 
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.28918 -0.41836 -0.03916  0.57284  1.29631 
## 
## Random effects:
##  Groups Name        Variance Std.Dev.
##  g      (Intercept) 0.08345  0.2889  
## Number of obs: 30, groups:  g, 3
## 
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  1.54079    0.27173   5.670 1.43e-08 ***
## x           -0.05000    0.03955  -1.264    0.206    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##   (Intr)
## x -0.666

## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula: z ~ x + (1 | g)
##    Data: simdata
## 
##      AIC      BIC   logLik deviance df.resid 
##    109.0    113.2    -51.5    103.0       27 
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.28918 -0.41836 -0.03916  0.57284  1.29631 
## 
## Random effects:
##  Groups Name        Variance Std.Dev.
##  g      (Intercept) 0.08345  0.2889  
## Number of obs: 30, groups:  g, 3
## 
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  1.54079    0.27173   5.670 1.43e-08 ***
## x           -0.05000    0.03955  -1.264    0.206    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##   (Intr)
## x -0.666

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## Power for predictor 'x', (95% confidence interval):
##       100.0% (83.16, 100.0)
## 
## Test: z-test
##       Effect size for x is -0.050
## 
## Based on 20 simulations, (0 warnings, 0 errors)
## alpha = 0.05, nrow = 150
## 
## Time elapsed: 0 h 0 m 0 s

The extended model2 has x values from 1 to 20, in three groups as before, for a total of 60 rows (compared to 30 in model1).

With observations at 20 values of x, we would have about 96% power to detect an effect of size −0·05.The study is clearly overpowered with that sample size.

Power analysis at a range of sample sizes

Next we use powerCurve function in simr to explore trade-offs between sample size and power.

We noted above that the study was overpowered when observations were taken at 20 values of the variable x. Could we reduce that number while keeping our power above the usual 80% threshold?

pc2 <- powerCurve(model2)

## Calculating power at 10 sample sizes along x

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## ( 1/10) Simulating: |=====================                                     |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |======================                                    |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=======================                                   |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |========================                                  |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=========================                                 |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |==========================                                |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |===========================                               |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |============================                              |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=============================                             |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |==============================                            |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |===============================                           |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |================================                          |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=================================                         |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |==================================                        |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |===================================                       |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |====================================                      |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=====================================                     |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |======================================                    |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=======================================                   |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |========================================                  |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=========================================                 |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |==========================================                |

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## ( 1/10) Simulating: |===========================================               |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |============================================              |

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## ( 1/10) Simulating: |=============================================             |

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## ( 1/10) Simulating: |==============================================            |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |===============================================           |

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## ( 1/10) Simulating: |================================================          |

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## ( 1/10) Simulating: |=================================================         |

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## ( 1/10) Simulating: |==================================================        |

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## ( 1/10) Simulating: |===================================================       |

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## ( 1/10) Simulating: |====================================================      |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=====================================================     |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |======================================================    |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |=======================================================   |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |========================================================  |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |========================================================= |

## boundary (singular) fit: see help('isSingular')
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## ( 1/10) Simulating: |==========================================================|( 1/10) ( 2/10) ( 2/10) Simulating: |                                                          |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=                                                         |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |==                                                        |

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## ( 2/10) Simulating: |===                                                       |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |====                                                      |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=====                                                     |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |======                                                    |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=======                                                   |

## boundary (singular) fit: see help('isSingular')

## ( 2/10) Simulating: |========                                                  |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=========                                                 |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |==========                                                |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |===========                                               |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |============                                              |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=============                                             |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |==============                                            |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |===============                                           |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |================                                          |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=================                                         |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |==================                                        |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |===================                                       |

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## ( 2/10) Simulating: |====================                                      |

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## ( 2/10) Simulating: |=====================                                     |

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## ( 2/10) Simulating: |======================                                    |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=======================                                   |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |========================                                  |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=========================                                 |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |==========================                                |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |===========================                               |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |============================                              |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=============================                             |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |==============================                            |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |===============================                           |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |================================                          |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |=================================                         |

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## ( 2/10) Simulating: |==================================                        |

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## ( 2/10) Simulating: |===================================                       |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |====================================                      |

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## ( 2/10) Simulating: |=====================================                     |

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## ( 2/10) Simulating: |======================================                    |

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## ( 2/10) Simulating: |=======================================                   |

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## ( 2/10) Simulating: |========================================                  |

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## ( 2/10) Simulating: |=========================================                 |

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## ( 2/10) Simulating: |==========================================                |

## boundary (singular) fit: see help('isSingular')
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## ( 2/10) Simulating: |===========================================               |

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## ( 2/10) Simulating: |============================================              |

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## ( 2/10) Simulating: |=============================================             |

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## ( 2/10) Simulating: |==============================================            |

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## ( 2/10) Simulating: |===============================================           |

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## ( 2/10) Simulating: |================================================          |

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## ( 2/10) Simulating: |=================================================         |

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## ( 2/10) Simulating: |==================================================        |

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## ( 2/10) Simulating: |===================================================       |

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## ( 2/10) Simulating: |====================================================      |

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## ( 2/10) Simulating: |=====================================================     |

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## ( 2/10) Simulating: |======================================================    |

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## ( 2/10) Simulating: |=======================================================   |

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## ( 2/10) Simulating: |========================================================  |

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## ( 2/10) Simulating: |========================================================= |

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## ( 2/10) Simulating: |==========================================================|( 2/10) ( 3/10) ( 3/10) Simulating: |                                                          |

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## ( 3/10) Simulating: |=                                                         |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |==                                                        |

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## ( 3/10) Simulating: |===                                                       |

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## ( 3/10) Simulating: |====                                                      |

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## ( 3/10) Simulating: |=====                                                     |

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## ( 3/10) Simulating: |======                                                    |

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## ( 3/10) Simulating: |=======                                                   |

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## ( 3/10) Simulating: |========                                                  |

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## ( 3/10) Simulating: |=========                                                 |

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## ( 3/10) Simulating: |==========                                                |

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## ( 3/10) Simulating: |===========                                               |

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## ( 3/10) Simulating: |============                                              |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |=============                                             |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |==============                                            |

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## ( 3/10) Simulating: |===============                                           |

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## ( 3/10) Simulating: |================                                          |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |=================                                         |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |==================                                        |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |===================                                       |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |====================                                      |

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## ( 3/10) Simulating: |=====================                                     |

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## ( 3/10) Simulating: |======================                                    |

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## ( 3/10) Simulating: |=======================                                   |

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## ( 3/10) Simulating: |========================                                  |

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## ( 3/10) Simulating: |=========================                                 |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |==========================                                |

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## ( 3/10) Simulating: |===========================                               |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |============================                              |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |=============================                             |

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## ( 3/10) Simulating: |==============================                            |

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## ( 3/10) Simulating: |===============================                           |

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## ( 3/10) Simulating: |================================                          |

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## ( 3/10) Simulating: |=================================                         |

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## ( 3/10) Simulating: |==================================                        |

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## ( 3/10) Simulating: |===================================                       |

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## ( 3/10) Simulating: |====================================                      |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |=====================================                     |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |======================================                    |

## boundary (singular) fit: see help('isSingular')
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## ( 3/10) Simulating: |=======================================                   |

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## ( 3/10) Simulating: |========================================                  |

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## ( 3/10) Simulating: |==========================================                |

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## ( 3/10) Simulating: |=============================================             |

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## ( 3/10) Simulating: |===============================================           |

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## ( 3/10) Simulating: |================================================          |

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## ( 3/10) Simulating: |=================================================         |

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## ( 3/10) Simulating: |==================================================        |

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## ( 3/10) Simulating: |===================================================       |

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## ( 3/10) Simulating: |====================================================      |

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## ( 3/10) Simulating: |======================================================    |

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## ( 3/10) Simulating: |=======================================================   |

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## ( 3/10) Simulating: |========================================================  |

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## ( 3/10) Simulating: |========================================================= |

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## ( 3/10) Simulating: |==========================================================|( 3/10) ( 4/10) ( 4/10) Simulating: |                                                          |

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## ( 4/10) Simulating: |=                                                         |

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## ( 4/10) Simulating: |==                                                        |

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## ( 4/10) Simulating: |===                                                       |

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## ( 4/10) Simulating: |====                                                      |

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## ( 4/10) Simulating: |=====                                                     |

## boundary (singular) fit: see help('isSingular')
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## ( 4/10) Simulating: |======                                                    |

## boundary (singular) fit: see help('isSingular')
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## ( 4/10) Simulating: |=======                                                   |

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## ( 4/10) Simulating: |========                                                  |

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## ( 4/10) Simulating: |=========                                                 |

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## ( 4/10) Simulating: |==========                                                |

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## ( 4/10) Simulating: |===========                                               |

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## ( 4/10) Simulating: |============                                              |

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## ( 4/10) Simulating: |=============                                             |

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## ( 4/10) Simulating: |==============                                            |

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## ( 4/10) Simulating: |===============                                           |

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## ( 4/10) Simulating: |================                                          |

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## ( 4/10) Simulating: |=================                                         |

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## ( 4/10) Simulating: |==================                                        |

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## ( 4/10) Simulating: |===================                                       |

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## ( 4/10) Simulating: |====================                                      |

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## ( 4/10) Simulating: |=====================                                     |

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## ( 4/10) Simulating: |======================                                    |

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## ( 4/10) Simulating: |=======================                                   |

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## ( 4/10) Simulating: |========================                                  |

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## ( 4/10) Simulating: |=========================                                 |

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## ( 4/10) Simulating: |==========================                                |

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## ( 4/10) Simulating: |===========================                               |

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## ( 4/10) Simulating: |============================                              |

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## ( 4/10) Simulating: |=============================                             |

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## ( 4/10) Simulating: |==============================                            |

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## ( 4/10) Simulating: |===============================                           |

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## ( 4/10) Simulating: |================================                          |

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## ( 4/10) Simulating: |=================================                         |

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## ( 4/10) Simulating: |==================================                        |

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## ( 4/10) Simulating: |===================================                       |

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## ( 4/10) Simulating: |====================================                      |

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## ( 4/10) Simulating: |=====================================                     |

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## ( 4/10) Simulating: |======================================                    |

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## ( 4/10) Simulating: |=======================================                   |

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## ( 4/10) Simulating: |========================================                  |

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## ( 4/10) Simulating: |=========================================                 |

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## ( 4/10) Simulating: |==============================================            |

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## ( 4/10) Simulating: |===============================================           |

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## ( 4/10) Simulating: |================================================          |

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## ( 4/10) Simulating: |=================================================         |

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## ( 4/10) Simulating: |===================================================       |( 4/10) Simulating: |====================================================      |

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## ( 4/10) Simulating: |========================================================  |

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## ( 4/10) Simulating: |========================================================= |

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## ( 4/10) Simulating: |==========================================================|( 4/10) ( 5/10) ( 5/10) Simulating: |                                                          |

## boundary (singular) fit: see help('isSingular')
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## ( 5/10) Simulating: |=                                                         |

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## ( 5/10) Simulating: |==                                                        |

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## ( 5/10) Simulating: |===                                                       |

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## ( 5/10) Simulating: |====                                                      |

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## ( 5/10) Simulating: |=====                                                     |

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## ( 5/10) Simulating: |======                                                    |

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## ( 5/10) Simulating: |=======                                                   |

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## ( 5/10) Simulating: |========                                                  |

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## ( 5/10) Simulating: |=========                                                 |

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## ( 5/10) Simulating: |==========                                                |

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## ( 5/10) Simulating: |===========                                               |

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## ( 5/10) Simulating: |============                                              |

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## ( 5/10) Simulating: |=============                                             |

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## ( 5/10) Simulating: |==============                                            |

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## ( 5/10) Simulating: |===============                                           |

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## ( 5/10) Simulating: |================                                          |

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## ( 5/10) Simulating: |=================                                         |

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## ( 5/10) Simulating: |==================                                        |

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## ( 5/10) Simulating: |===================                                       |

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## ( 5/10) Simulating: |====================                                      |

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## ( 5/10) Simulating: |=====================                                     |

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## ( 5/10) Simulating: |======================                                    |

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## ( 5/10) Simulating: |=======================                                   |

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## ( 5/10) Simulating: |=========================                                 |

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## ( 5/10) Simulating: |============================                              |

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## ( 5/10) Simulating: |===============================                           |

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## ( 5/10) Simulating: |===================================================       |( 5/10) Simulating: |====================================================      |

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## ( 5/10) Simulating: |========================================================= |

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## ( 5/10) Simulating: |==========================================================|( 5/10) ( 6/10) ( 6/10) Simulating: |                                                          |

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## ( 6/10) Simulating: |=                                                         |

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## ( 6/10) Simulating: |==                                                        |

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## ( 6/10) Simulating: |===                                                       |

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## ( 6/10) Simulating: |====                                                      |

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## ( 6/10) Simulating: |=====                                                     |

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## ( 6/10) Simulating: |======                                                    |

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## ( 6/10) Simulating: |=======                                                   |

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## ( 6/10) Simulating: |========                                                  |

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## ( 6/10) Simulating: |=========                                                 |

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## ( 6/10) Simulating: |==========                                                |

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## ( 6/10) Simulating: |===========                                               |

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## ( 6/10) Simulating: |============                                              |

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## ( 6/10) Simulating: |=============                                             |

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## ( 6/10) Simulating: |==============                                            |

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## ( 6/10) Simulating: |===============                                           |

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## ( 6/10) Simulating: |================                                          |

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## ( 6/10) Simulating: |=================                                         |

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## ( 6/10) Simulating: |==================                                        |

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## ( 6/10) Simulating: |===================                                       |

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## ( 6/10) Simulating: |====================                                      |

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## ( 6/10) Simulating: |=====================                                     |

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## ( 6/10) Simulating: |=======================                                   |

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## ( 6/10) Simulating: |============================                              |

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## ( 6/10) Simulating: |=================================                         |

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## ( 6/10) Simulating: |===================================================       |( 6/10) Simulating: |====================================================      |

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## ( 6/10) Simulating: |==========================================================|( 6/10) ( 7/10) ( 7/10) Simulating: |                                                          |

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## ( 7/10) Simulating: |=                                                         |

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## ( 7/10) Simulating: |==                                                        |

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## ( 7/10) Simulating: |===                                                       |

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## ( 7/10) Simulating: |====                                                      |

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## ( 7/10) Simulating: |=====                                                     |

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## ( 7/10) Simulating: |======                                                    |

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## ( 7/10) Simulating: |=======                                                   |

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## ( 7/10) Simulating: |========                                                  |

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## ( 7/10) Simulating: |=========                                                 |

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## ( 7/10) Simulating: |==========                                                |

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## ( 7/10) Simulating: |===========                                               |

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## ( 7/10) Simulating: |============                                              |

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## ( 7/10) Simulating: |=============                                             |

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## ( 7/10) Simulating: |=================                                         |

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## ( 7/10) Simulating: |==================                                        |

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## ( 7/10) Simulating: |===================                                       |

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## ( 7/10) Simulating: |====================                                      |

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## ( 7/10) Simulating: |=====================                                     |

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## ( 7/10) Simulating: |=======================                                   |

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## ( 7/10) Simulating: |============================                              |

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## ( 7/10) Simulating: |=================================                         |

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## ( 7/10) Simulating: |===================================================       |( 7/10) Simulating: |====================================================      |

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## ( 7/10) Simulating: |========================================================= |

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## ( 7/10) Simulating: |==========================================================|( 7/10) ( 8/10) ( 8/10) Simulating: |                                                          |

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## ( 8/10) Simulating: |=                                                         |

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## ( 8/10) Simulating: |==                                                        |

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## ( 8/10) Simulating: |===                                                       |

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## ( 8/10) Simulating: |====                                                      |

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## ( 8/10) Simulating: |=====                                                     |

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## ( 8/10) Simulating: |======                                                    |

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## ( 8/10) Simulating: |=======                                                   |

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## ( 8/10) Simulating: |========                                                  |

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## ( 8/10) Simulating: |=========                                                 |

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## ( 8/10) Simulating: |==========                                                |

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## ( 8/10) Simulating: |===========                                               |

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## ( 8/10) Simulating: |============                                              |

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## ( 8/10) Simulating: |==============                                            |

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## ( 8/10) Simulating: |===============                                           |

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## ( 8/10) Simulating: |================                                          |

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## ( 8/10) Simulating: |=================                                         |

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## ( 8/10) Simulating: |===================                                       |

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## ( 8/10) Simulating: |====================                                      |

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## ( 8/10) Simulating: |=====================                                     |

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## ( 8/10) Simulating: |============================                              |

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## ( 8/10) Simulating: |===================================================       |( 8/10) Simulating: |====================================================      |

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## ( 8/10) Simulating: |==========================================================|( 8/10) ( 9/10) ( 9/10) Simulating: |                                                          |

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## ( 9/10) Simulating: |=                                                         |

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## ( 9/10) Simulating: |==                                                        |

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## ( 9/10) Simulating: |===                                                       |

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## ( 9/10) Simulating: |====                                                      |

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## ( 9/10) Simulating: |=====                                                     |

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## ( 9/10) Simulating: |======                                                    |

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## ( 9/10) Simulating: |=======                                                   |

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## ( 9/10) Simulating: |========                                                  |

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## ( 9/10) Simulating: |=========                                                 |

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## ( 9/10) Simulating: |==========                                                |

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## ( 9/10) Simulating: |===========                                               |

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## ( 9/10) Simulating: |============                                              |

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## ( 9/10) Simulating: |==============                                            |

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## ( 9/10) Simulating: |===============                                           |

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## ( 9/10) Simulating: |================                                          |

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## ( 9/10) Simulating: |=================                                         |

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## ( 9/10) Simulating: |==================                                        |

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## ( 9/10) Simulating: |===================                                       |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |====================                                      |

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## ( 9/10) Simulating: |=====================                                     |

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## ( 9/10) Simulating: |======================                                    |

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## ( 9/10) Simulating: |=======================                                   |

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## ( 9/10) Simulating: |========================                                  |

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## ( 9/10) Simulating: |=========================                                 |

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## ( 9/10) Simulating: |==========================                                |

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## ( 9/10) Simulating: |===========================                               |

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## ( 9/10) Simulating: |============================                              |

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## ( 9/10) Simulating: |=============================                             |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |==============================                            |

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## ( 9/10) Simulating: |===============================                           |

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## ( 9/10) Simulating: |================================                          |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |=================================                         |

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## ( 9/10) Simulating: |==================================                        |

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## ( 9/10) Simulating: |===================================                       |

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## ( 9/10) Simulating: |====================================                      |

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## ( 9/10) Simulating: |======================================                    |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |=======================================                   |

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## ( 9/10) Simulating: |========================================                  |

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## ( 9/10) Simulating: |=========================================                 |

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## ( 9/10) Simulating: |==========================================                |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |===========================================               |

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## ( 9/10) Simulating: |============================================              |

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## ( 9/10) Simulating: |=============================================             |

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## ( 9/10) Simulating: |==============================================            |

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## ( 9/10) Simulating: |===============================================           |

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## ( 9/10) Simulating: |================================================          |

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## ( 9/10) Simulating: |=================================================         |

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## ( 9/10) Simulating: |==================================================        |

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## ( 9/10) Simulating: |===================================================       |( 9/10) Simulating: |====================================================      |

## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |=====================================================     |

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## ( 9/10) Simulating: |======================================================    |

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## ( 9/10) Simulating: |=======================================================   |

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## ( 9/10) Simulating: |========================================================  |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |========================================================= |

## boundary (singular) fit: see help('isSingular')
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## ( 9/10) Simulating: |==========================================================|( 9/10) (10/10) (10/10) Simulating: |                                                          |

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## (10/10) Simulating: |=                                                         |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |==                                                        |

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## (10/10) Simulating: |===                                                       |

## boundary (singular) fit: see help('isSingular')

## (10/10) Simulating: |====                                                      |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=====                                                     |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |======                                                    |

## boundary (singular) fit: see help('isSingular')
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## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=======                                                   |

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## (10/10) Simulating: |========                                                  |

## boundary (singular) fit: see help('isSingular')

## (10/10) Simulating: |=========                                                 |

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## (10/10) Simulating: |==========                                                |

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## (10/10) Simulating: |===========                                               |

## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |============                                              |

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## (10/10) Simulating: |=============                                             |

## boundary (singular) fit: see help('isSingular')

## (10/10) Simulating: |==============                                            |

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## (10/10) Simulating: |===============                                           |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |================                                          |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=================                                         |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |==================                                        |

## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |===================                                       |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |====================                                      |

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## (10/10) Simulating: |=====================                                     |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |======================                                    |

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## (10/10) Simulating: |=======================                                   |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=========================                                 |

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## (10/10) Simulating: |===========================                               |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |============================                              |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=============================                             |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |==============================                            |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |================================                          |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=================================                         |

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## (10/10) Simulating: |======================================                    |

## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=======================================                   |

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## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |===========================================               |

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## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |================================================          |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |=================================================         |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |==================================================        |

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## (10/10) Simulating: |===================================================       |(10/10) Simulating: |====================================================      |

## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |========================================================  |

## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
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## (10/10) Simulating: |========================================================= |

## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')

## (10/10) Simulating: |==========================================================|(10/10) 

print(pc2)

## Power for predictor 'x', (95% confidence interval),
## by largest value of x:
##       3:  6.50% ( 5.05,  8.21) - 9 rows
##       8: 20.60% (18.13, 23.24) - 24 rows
##      13: 58.20% (55.07, 61.28) - 39 rows
##      19: 93.30% (91.57, 94.77) - 57 rows
##      24: 99.40% (98.70, 99.78) - 72 rows
##      29: 100.0% (99.63, 100.0) - 87 rows
##      34: 100.0% (99.63, 100.0) - 102 rows
##      40: 100.0% (99.63, 100.0) - 120 rows
##      45: 100.0% (99.63, 100.0) - 135 rows
##      50: 100.0% (99.63, 100.0) - 150 rows
## 
## Time elapsed: 0 h 4 m 10 s

plot(pc2)

Increasing the size within groups

We can replace the along argument to extend and powerCurve with the within argument to increase the sample size within groups. Each group has only one observation at each level of x and g. We can extend this to five observations per site per year as follows:

model4 <- extend(model1, within="x+g", n=5)

pc4 <- powerCurve(model4, within="x+g", breaks=1:5, nsim=20)

## Calculating power at 5 sample sizes within x+g

## Simulating: |                                                                  |Simulating: |===                                                               |Simulating: |======                                                            |Simulating: |=========                                                         |Simulating: |=============                                                     |Simulating: |================                                                  |Simulating: |===================                                               |Simulating: |=======================                                           |Simulating: |==========================                                        |Simulating: |=============================                                     |Simulating: |=================================                                 |Simulating: |====================================                              |Simulating: |=======================================                           |Simulating: |==========================================                        |Simulating: |==============================================                    |Simulating: |=================================================                 |Simulating: |====================================================              |Simulating: |========================================================          |Simulating: |===========================================================       |Simulating: |==============================================================    |Simulating: |==================================================================|(1/5) (1/5) Simulating: |                                                            |

## boundary (singular) fit: see help('isSingular')

## (1/5) Simulating: |===                                                         |

## boundary (singular) fit: see help('isSingular')

## (1/5) Simulating: |======                                                      |

## boundary (singular) fit: see help('isSingular')

## (1/5) Simulating: |=========                                                   |(1/5) Simulating: |============                                                |(1/5) Simulating: |===============                                             |(1/5) Simulating: |==================                                          |(1/5) Simulating: |=====================                                       |(1/5) Simulating: |========================                                    |(1/5) Simulating: |===========================                                 |(1/5) Simulating: |==============================                              |(1/5) Simulating: |=================================                           |(1/5) Simulating: |====================================                        |(1/5) Simulating: |=======================================                     |(1/5) Simulating: |==========================================                  |(1/5) Simulating: |=============================================               |

## boundary (singular) fit: see help('isSingular')

## (1/5) Simulating: |================================================            |

## boundary (singular) fit: see help('isSingular')

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## boundary (singular) fit: see help('isSingular')

## (1/5) Simulating: |======================================================      |(1/5) Simulating: |=========================================================   |(1/5) Simulating: |============================================================|(1/5) (2/5) (2/5) Simulating: |                                                            |

## boundary (singular) fit: see help('isSingular')

## (2/5) Simulating: |===                                                         |

## boundary (singular) fit: see help('isSingular')

## (2/5) Simulating: |======                                                      |(2/5) Simulating: |=========                                                   |(2/5) Simulating: |============                                                |(2/5) Simulating: |===============                                             |(2/5) Simulating: |==================                                          |(2/5) Simulating: |=====================                                       |(2/5) Simulating: |========================                                    |(2/5) Simulating: |===========================                                 |(2/5) Simulating: |==============================                              |(2/5) Simulating: |=================================                           |(2/5) Simulating: |====================================                        |(2/5) Simulating: |=======================================                     |(2/5) Simulating: |==========================================                  |(2/5) Simulating: |=============================================               |(2/5) Simulating: |================================================            |

## boundary (singular) fit: see help('isSingular')

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## boundary (singular) fit: see help('isSingular')

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## boundary (singular) fit: see help('isSingular')

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## boundary (singular) fit: see help('isSingular')

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## boundary (singular) fit: see help('isSingular')

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print(pc4)

## Power for predictor 'x', (95% confidence interval),
## by number of observations within x+g:
##       1: 35.00% (15.39, 59.22) - 30 rows
##       2: 50.00% (27.20, 72.80) - 60 rows
##       3: 75.00% (50.90, 91.34) - 90 rows
##       4: 80.00% (56.34, 94.27) - 120 rows
##       5: 90.00% (68.30, 98.77) - 150 rows
## 
## Time elapsed: 0 h 0 m 2 s

Variance Partion Coefficient VPC or ICC

The purpose of multilevel models is to partition variance in the outcome between the different groupings in the data. We want to know what proportion of the total variance is attributable to variation within-hospitals/groups, or how much is found between-groups/hopsitals. This statistic is termed the variance partition coefficient VPC, or intraclass correlation.

Example We calculate the VPC with some simple arithmetic on the variance estimates from the lmer model. We can extract the variance estimates from the VarCorr function:

Note: We convert the result of VarCorr to a dataframe, we are provided with the columns vcov which stands for variance or covariance, as well as the sdcor (standard deviation or correlation) which is provided in the printed summary:

random.intercepts.model <- lmer(Reaction ~ Days + (1|Subject),  data=lme4::sleepstudy)
VarCorr(random.intercepts.model)

##  Groups   Name        Std.Dev.
##  Subject  (Intercept) 37.124  
##  Residual             30.991

rand(random.intercepts.model)

## ANOVA-like table for random-effects: Single term deletions
## 
## Model:
## Reaction ~ Days + (1 | Subject)
##               npar  logLik    AIC   LRT Df Pr(>Chisq)    
## <none>           4 -893.23 1794.5                        
## (1 | Subject)    3 -946.83 1899.7 107.2  1  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The variance partition coefficient is simply the variance at a given level of the model, divided by the total variance (the sum of the variance parameters).

VarCorr(random.intercepts.model) %>%
  as_data_frame() %>%
  mutate(icc=vcov/sum(vcov)) %>%
  select(grp, icc)

## Warning: `as_data_frame()` was deprecated in tibble 2.0.0.
## Please use `as_tibble()` instead.
## The signature and semantics have changed, see `?as_tibble`.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.

## # A tibble: 2 × 2
##   grp        icc
##   <chr>    <dbl>
## 1 Subject  0.589
## 2 Residual 0.411

Conclusion Intraclass correlations were computed from the mixed effects mode. 59% of the variation in outcome was attributable to differences between subjects, \(\chi^{2}_{1}=107.2\), p<.001.

Setting variance parameters to represent clustering within groups

Requires setting one of the variance parameters of our model to represent clustering within group. 1. First we can use VarCorr() to check the variance parameters of the model we just ran:

VarCorr(model1)

##  Groups Name        Std.Dev.
##  g      (Intercept) 0.28888

2. Then we could simulate increasing the variance parameter for g to 0.5:

VarCorr(model1)['g'] <- .5
powerSim(model1, nsim=10)

## Simulating: |                                                                  |

## boundary (singular) fit: see help('isSingular')

## Simulating: |======                                                            |Simulating: |=============                                                     |

## boundary (singular) fit: see help('isSingular')

## Simulating: |===================                                               |Simulating: |==========================                                        |Simulating: |=================================                                 |Simulating: |=======================================                           |Simulating: |==============================================                    |Simulating: |====================================================              |Simulating: |===========================================================       |Simulating: |==================================================================|

## Power for predictor 'x', (95% confidence interval):
##       50.00% (18.71, 81.29)
## 
## Test: z-test
##       Effect size for x is -0.050
## 
## Based on 10 simulations, (0 warnings, 0 errors)
## alpha = 0.05, nrow = 30
## 
## Time elapsed: 0 h 0 m 0 s

Note As the clustering in our data increases our statistical power goes down. This is because, when clustering is present, each new observation (row) in the dataset provides less new information to estimate our treatment effect.

Note In this example we increased the variance associated with the group by quite a lot: setting the variance of g to 0.5 equates to an ICC for g of .33 ( .5 / (.5 + 1) = .33).

Varying the number and size of groups

We can add extra levels for groups the same way we added extra values for x. For example if the variable g represents our study sites, we could increase the number of sites from 3 to 20.

VarCorr(model1)['g'] <- .5
model3 <- extend(model1, along="g", n=20)
summary(model3)

## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula: z ~ x + (1 | g)
##    Data: simdata
## 
##      AIC      BIC   logLik deviance df.resid 
##    109.0    113.2    -51.5    103.0       27 
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.28918 -0.41836 -0.03916  0.57284  1.29631 
## 
## Random effects:
##  Groups Name        Variance Std.Dev.
##  g      (Intercept) 0.5      0.7071  
## Number of obs: 30, groups:  g, 3
## 
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  1.54079    0.27173   5.670 1.43e-08 ***
## x           -0.05000    0.03955  -1.264    0.206    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##   (Intr)
## x -0.666

pc3 <- powerCurve(model3, along="g", nsim=25)

## Calculating power at 10 sample sizes along g

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plot(pc3) 

Different tests

1. Refer to the vignette on hypothesis tests using do test option in simr for details of different terst options.

2. For illustration, lets say we are intrested in testing single random/cetnre effect

model1 <- glmer(z ~ x + (1|g), family="poisson", data=simdata) # no need to rerun its already there
summary(model1)

## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula: z ~ x + (1 | g)
##    Data: simdata
## 
##      AIC      BIC   logLik deviance df.resid 
##    109.0    113.2    -51.5    103.0       27 
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.28918 -0.41836 -0.03916  0.57284  1.29631 
## 
## Random effects:
##  Groups Name        Variance Std.Dev.
##  g      (Intercept) 0.08345  0.2889  
## Number of obs: 30, groups:  g, 3
## 
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  1.54079    0.27173   5.670 1.43e-08 ***
## x           -0.11481    0.03955  -2.903   0.0037 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##   (Intr)
## x -0.666

doTest(model1)

## p-value for predictor 'x': 0.003698474
##           --------------------
## Test: z-test
##       Effect size for x is -0.11

Side note to convert a model to LaTeX, feed a model object to extract_eq():

\[ \operatorname{mpg} = \alpha + \beta_{1}(\operatorname{cyl}) + \beta_{2}(\operatorname{disp}) + \epsilon \]

# Fit a simple model
mod1 <- lm(mpg ~ cyl + disp, mtcars)

# Give the results to extract_eq
extract_eq(mod1) 

\[ \operatorname{mpg} = \alpha + \beta_{1}(\operatorname{cyl}) + \beta_{2}(\operatorname{disp}) + \epsilon \]

When using categorical variables, it will include the levels of the variables as subscripts. Here, we use data from the {palmerpenguins} dataset. \[ \operatorname{body\_mass\_g} = \alpha + \beta_{1}(\operatorname{bill\_length\_mm}) + \beta_{2}(\operatorname{species}_{\operatorname{Chinstrap}}) + \beta_{3}(\operatorname{species}_{\operatorname{Gentoo}}) + \epsilon \]

mod3 <- lm(body_mass_g ~ bill_length_mm + species, penguins)
extract_eq(mod3)

\[ \operatorname{body\_mass\_g} = \alpha + \beta_{1}(\operatorname{bill\_length\_mm}) + \beta_{2}(\operatorname{species}_{\operatorname{Chinstrap}}) + \beta_{3}(\operatorname{species}_{\operatorname{Gentoo}}) + \epsilon \]