Analysis of Covariance

Analysis of covariance

  • ANOVA: explanatory variables categorical (divide data into groups)

  • traditionally, analysis of covariance has categorical \(x\)’s plus one numerical \(x\) (“covariate”) to be adjusted for.

  • lm handles this too.

  • Simple example: two treatments (drugs) (a and b), with before and after scores.

  • Does knowing before score and/or treatment help to predict after score?

  • Is after score different by treatment/before score?

Data

Treatment, before, after:

a 5 20
a 10 23
a 12 30
a 9 25
a 23 34
a 21 40
a 14 27
a 18 38
a 6 24
a 13 31
b 7 19
b 12 26
b 27 33
b 24 35
b 18 30
b 22 31
b 26 34
b 21 28
b 14 23
b 9 22

Packages

library(tidyverse)
library(broom)
library(marginaleffects)

the last of these for predictions.

Read in data

url <- "http://ritsokiguess.site/datafiles/ancova.txt"
prepost <- read_delim(url, " ")
prepost

Making a plot

ggplot(prepost, aes(x = before, y = after, colour = drug)) +
  geom_point() + geom_smooth(method = "lm")

Comments

  • As before score goes up, after score goes up.

  • Red points (drug A) generally above blue points (drug B), for comparable before score.

  • Suggests before score effect and drug effect.

The means

prepost %>%
  group_by(drug) %>%
  summarize(
    before_mean = mean(before),
    after_mean = mean(after)
  )

  • Mean “after” score slightly higher for treatment A.

  • Mean “before” score much higher for treatment B.

  • Greater improvement on treatment A.

Testing for interaction

prepost.1 <- lm(after ~ before * drug, data = prepost)
anova(prepost.1)
summary(prepost.1)

Call:
lm(formula = after ~ before * drug, data = prepost)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.8562 -1.7500  0.0696  1.8982  4.0207 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   16.4226     2.0674   7.944 6.08e-07 ***
before         0.9754     0.1446   6.747 4.69e-06 ***
drugb         -1.3139     3.1310  -0.420    0.680    
before:drugb  -0.2536     0.1893  -1.340    0.199    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.622 on 16 degrees of freedom
Multiple R-squared:  0.8355,    Adjusted R-squared:  0.8046 
F-statistic: 27.09 on 3 and 16 DF,  p-value: 1.655e-06
  • Interaction not significant. Will remove later.

Predictions

Set up values to predict for:

summary(prepost)
     drug               before          after      
 Length:20          Min.   : 5.00   Min.   :19.00  
 Class :character   1st Qu.: 9.75   1st Qu.:23.75  
 Mode  :character   Median :14.00   Median :29.00  
                    Mean   :15.55   Mean   :28.65  
                    3rd Qu.:21.25   3rd Qu.:33.25  
                    Max.   :27.00   Max.   :40.00  
new <- datagrid(before = c(9.75, 14, 21.25), 
                drug = c("a", "b"), model = prepost.1)
new

and then

cbind(predictions(prepost.1, newdata = new)) %>% 
  select(drug, before, estimate, conf.low, conf.high)

Predictions (with interaction included), plotted

plot_predictions(model = prepost.1, condition = c("before", "drug"))

Lines almost parallel, but not quite.

Taking out interaction

prepost.2 <- update(prepost.1, . ~ . - before:drug)
summary(prepost.2)

Call:
lm(formula = after ~ before + drug, data = prepost)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.6348 -2.5099 -0.2038  1.8871  4.7453 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  18.3600     1.5115  12.147 8.35e-10 ***
before        0.8275     0.0955   8.665 1.21e-07 ***
drugb        -5.1547     1.2876  -4.003 0.000921 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.682 on 17 degrees of freedom
Multiple R-squared:  0.817, Adjusted R-squared:  0.7955 
F-statistic: 37.96 on 2 and 17 DF,  p-value: 5.372e-07
anova(prepost.2)

  • Take out non-significant interaction.

  • before and drug strongly significant.

  • Do predictions again and plot them.

Predictions

cbind(predictions(prepost.2, newdata = new)) %>% 
  select(drug, before, estimate)

Plot of predicted values

plot_predictions(prepost.2, condition = c("before", "drug"))

This time the lines are exactly parallel. No-interaction model forces them to have the same slope.

Different look at model output

  • anova(prepost.2) tests for significant effect of before score and of drug, but doesn’t help with interpretation.

  • summary(prepost.2) views as regression with slopes:

summary(prepost.2)

Call:
lm(formula = after ~ before + drug, data = prepost)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.6348 -2.5099 -0.2038  1.8871  4.7453 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  18.3600     1.5115  12.147 8.35e-10 ***
before        0.8275     0.0955   8.665 1.21e-07 ***
drugb        -5.1547     1.2876  -4.003 0.000921 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.682 on 17 degrees of freedom
Multiple R-squared:  0.817, Adjusted R-squared:  0.7955 
F-statistic: 37.96 on 2 and 17 DF,  p-value: 5.372e-07

Understanding those slopes

tidy(prepost.2)

  • before ordinary numerical variable; drug categorical.

  • lm uses first category druga as baseline.

  • Intercept is prediction of after score for before score 0 and drug A.

  • before slope is predicted change in after score when before score increases by 1 (usual slope)

  • Slope for drugb is change in predicted after score for being on drug B rather than drug A. Same for any before score (no interaction).

Summary

  • ANCOVA model: fits different regression line for each group, predicting response from covariate.

  • ANCOVA model with interaction between factor and covariate allows different slopes for each line.

  • Sometimes those lines can cross over!

  • If interaction not significant, take out. Lines then parallel.

  • With parallel lines, groups have consistent effect regardless of value of covariate.