Contrasts in single-cases

contrasts

regression

statistics

Author

Jürgen Wilbert

Published

September 27, 2022

Abstract

Contrasts in regression modells

Example dataset

plot
table

Single case data frame with one cases
Pawel
values	mt	phase
10	1	A1
19	2	A1
11	3	A1
6	4	A1
16	5	A1
16	6	A1
17	7	A1
18	8	A1
12	9	A1
12	10	A1
39	11	B1
18	12	B1
25	13	B1
31	14	B1
29	15	B1
24	16	B1
37	17	B1
17	18	B1
21	19	B1
18	20	B1
16	21	A2
17	22	A2
19	23	A2
21	24	A2
10	25	A2
11	26	A2
23	27	A2
11	28	A2
10	29	A2
9	30	A2
34	31	B2
35	32	B2
28	33	B2
30	34	B2
22	35	B2
21	36	B2
21	37	B2
22	38	B2
23	39	B2
34	40	B2

# mean of all phases

(means <- tapply(df$values, df$phase, mean))
  A1   B1   A2   B2 
13.7 25.9 14.7 27.0 
(grand_mean <- mean(means))
[1] 20.325

# Plm function reference:
plm(exampleA1B1A2B2$Pawel, contrast = "first", trend = FALSE, slope = FALSE) %>% coef() %>% .[,1]

(Intercept)     phaseB1     phaseA2     phaseB2 
       13.7        12.2         1.0        13.3

plm(exampleA1B1A2B2$Pawel, contrast = "preceding", trend = FALSE, slope = FALSE) %>% coef() %>% .[,1]

(Intercept)     phaseB1     phaseA2     phaseB2 
       13.7        12.2       -11.2        12.3

Contrasts

Treatment contrast

Compare mean of second to last phase against first phase (intercept)

The Intercept is the mean of a reference phase (defaults to the first).
The predictors are the differences from a phase to the reference (intercept).

treatment <- contr.treatment(4)
colnames(treatment) <- c("B1vsA1","A2vsA1","B2vsA1")
treatment

  B1vsA1 A2vsA1 B2vsA1
1      0      0      0
2      1      0      0
3      0      1      0
4      0      0      1

lm(values~phase, data=df, contrasts = list(phase = treatment)) |> coef()

(Intercept) phaseB1vsA1 phaseA2vsA1 phaseB2vsA1 
       13.7        12.2         1.0        13.3

c(Intercept = 13.7, # phaseA1
  phaseB1vsA1 = 25.9 - 13.7, 
  phaseA2vsA1 = 14.7 - 13.7,
  phaseB2vsA1 = 27 - 13.7)

  Intercept phaseB1vsA1 phaseA2vsA1 phaseB2vsA1 
       13.7        12.2         1.0        13.3

Sum contrast

Comparison from the first to the second last phase with the overall mean (Intercept is overall mean)

The Intercept is the overall mean.
The predictors are the differences from a phase mean to the overall mean.

sum <- contr.sum(4)
colnames(sum) <- c("A1vsMean","B1vsMean","A2vsMean")
sum

  A1vsMean B1vsMean A2vsMean
1        1        0        0
2        0        1        0
3        0        0        1
4       -1       -1       -1

lm(values~phase, data=df, contrasts = list(phase = sum)) |> coef()

  (Intercept) phaseA1vsMean phaseB1vsMean phaseA2vsMean 
       20.325        -6.625         5.575        -5.625

c(Intercept = grand_mean,
  phaseA1vsMean = 13.7 - grand_mean, 
  phaseB1vsMean = 25.9 - grand_mean, 
  phaseA2vsMean = 14.7 - grand_mean)

    Intercept phaseA1vsMean phaseB1vsMean phaseA2vsMean 
       20.325        -6.625         5.575        -5.625

Helmert contrast

Compares from the second to the last phase with the mean of all preceding phases.

The Intercept is the overall mean.
The predictors are the mean differences from a phase mean to the mean of all preceding phases.

helmert <- contr.helmert(4)
colnames(helmert) <- c("B1vsA1", "A2vsA1_B1", "B2vsA1_B1_A2")
helmert

  B1vsA1 A2vsA1_B1 B2vsA1_B1_A2
1     -1        -1           -1
2      1        -1           -1
3      0         2           -1
4      0         0            3

lm(values~phase, data=df, contrasts = list(phase = helmert)) |> coef()

      (Intercept)       phaseB1vsA1    phaseA2vsA1_B1 phaseB2vsA1_B1_A2 
           20.325             6.100            -1.700             2.225

c(Intercept = grand_mean,
  phaseB1vsA1 = (25.9 + -13.7) / 2, 
  phaseA2vsA1_B1 = (14.7  + (-13.7-25.9)/2) / 3,
  phaseB2vsA1_B1_A2 = (27 + (-13.7-25.9-14.7)/3) /4)

        Intercept       phaseB1vsA1    phaseA2vsA1_B1 phaseB2vsA1_B1_A2 
           20.325             6.100            -1.700             2.225

Revert Helmert

Comparing the second to last phase against the mean of all preceding phases

The Intercept is the grand mean.
The predictors are the differences from a phase to the mean of the preeding phases.

Compare phase2 vs. phase 1, phase 3 vs. mean of phase 1 and 2, phase4 vs. mean of phases 1,2,3 (Intercept is overall mean)

revers_helmert <- matrix(c(-1/2, 1/2, 0, 0, -1/3, -1/3, 2/3, 0, -1/4, -1/4, -1/4, 3/4), ncol = 3)
colnames(revers_helmert) <- c("B1vsA1", "A2vsA1_B1", "B2vsA1_B1_A2")
revers_helmert

     B1vsA1  A2vsA1_B1 B2vsA1_B1_A2
[1,]   -0.5 -0.3333333        -0.25
[2,]    0.5 -0.3333333        -0.25
[3,]    0.0  0.6666667        -0.25
[4,]    0.0  0.0000000         0.75

lm(values~phase, data=df, contrasts = list(phase = revers_helmert)) |> coef()

      (Intercept)       phaseB1vsA1    phaseA2vsA1_B1 phaseB2vsA1_B1_A2 
           20.325            12.200            -5.100             8.900

c(Intercept = grand_mean,
  phaseB1vsA1 = 25.7 - 13.7,
  phaseA2vsA1_B1 = 14.7 - (13.7+25.9)/2,
  phaseB2vsA1_B1_A2 = 27 - (13.7+25.9+14.7)/3)

        Intercept       phaseB1vsA1    phaseA2vsA1_B1 phaseB2vsA1_B1_A2 
           20.325            12.000            -5.100             8.900

Cummulative

Compare mean of second to last phase to mean of preceding phase

The Intercept is the mean of the first phase.
The predictors are the differences from a phase to the preeding phase.

cumulative <- matrix(c(0,1,1,1, 0,0,1,1, 0,0,0,1), ncol = 3)
colnames(cumulative) <- c("B1vsA1","A2vsB1","B2vsA2")
cumulative

     B1vsA1 A2vsB1 B2vsA2
[1,]      0      0      0
[2,]      1      0      0
[3,]      1      1      0
[4,]      1      1      1

lm(values~phase, data=df, contrasts = list(phase = cumulative)) |> coef()

(Intercept) phaseB1vsA1 phaseA2vsB1 phaseB2vsA2 
       13.7        12.2       -11.2        12.3

c(Intercept = 13.7,
  phaseB1vsA1 = 25.9 - 13.7, 
  phaseA2vsB1 = 14.7 - 25.9,
  phaseB2vsA2 = 27 - 14.7)

  Intercept phaseB1vsA1 phaseA2vsB1 phaseB2vsA2 
       13.7        12.2       -11.2        12.3

Contrast computation

treatment

(treatment <- contr.treatment(4))
  2 3 4
1 0 0 0
2 1 0 0
3 0 1 0
4 0 0 1
(treatment <-  cbind(1, treatment))
    2 3 4
1 1 0 0 0
2 1 1 0 0
3 1 0 1 0
4 1 0 0 1

(treatment <- solve(t(treatment)))
     2  3  4
1 1 -1 -1 -1
2 0  1  0  0
3 0  0  1  0
4 0  0  0  1
(treatment <- treatment[,-1])
   2  3  4
1 -1 -1 -1
2  1  0  0
3  0  1  0
4  0  0  1

colnames(treatment) <- c("B1vsA1","A2vsA1","B2vsA1")

lm(values~phase, data=df, contrasts = list(phase = treatment)) |> summary()

Call:
lm(formula = values ~ phase, data = df, contrasts = list(phase = treatment))

Residuals:
   Min     1Q Median     3Q    Max 
 -8.90  -4.75  -1.30   4.30  13.10 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  20.3250     0.9373  21.684  < 2e-16 ***
phaseB1vsA1   5.5750     1.6235   3.434 0.001513 ** 
phaseA2vsA1  -5.6250     1.6235  -3.465 0.001389 ** 
phaseB2vsA1   6.6750     1.6235   4.112 0.000217 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.928 on 36 degrees of freedom
Multiple R-squared:  0.5444,    Adjusted R-squared:  0.5064 
F-statistic: 14.34 on 3 and 36 DF,  p-value: 2.635e-06

sum

(sum <- contr.sum(4))

  [,1] [,2] [,3]
1    1    0    0
2    0    1    0
3    0    0    1
4   -1   -1   -1

(sum <-  cbind(1, sum))

  [,1] [,2] [,3] [,4]
1    1    1    0    0
2    1    0    1    0
3    1    0    0    1
4    1   -1   -1   -1

(sum <- solve(t(sum)))

  [,1]  [,2]  [,3]  [,4]
1 0.25  0.75 -0.25 -0.25
2 0.25 -0.25  0.75 -0.25
3 0.25 -0.25 -0.25  0.75
4 0.25 -0.25 -0.25 -0.25

(sum <- sum[,-1])

   [,1]  [,2]  [,3]
1  0.75 -0.25 -0.25
2 -0.25  0.75 -0.25
3 -0.25 -0.25  0.75
4 -0.25 -0.25 -0.25

colnames(sum) <- c("A1vsMean","B1vsMean","A2vsMean")


lm(values~phase, data=df, contrasts = list(phase = sum)) |> coef()

  (Intercept) phaseA1vsMean phaseB1vsMean phaseA2vsMean 
       20.325       -13.300        -1.100       -12.300

c(Intercept = grand_mean,
  phaseA1vsMean = 13.7 - grand_mean, 
  phaseB1vsMean = 25.9 - grand_mean, 
  phaseA2vsMean = 14.7 - grand_mean)

    Intercept phaseA1vsMean phaseB1vsMean phaseA2vsMean 
       20.325        -6.625         5.575        -5.625