Designs with Two Factors

A Linear Model

Babies Walking

A linear model

m1 <- lm(walk ~ program, data = babies)
summary(m1)

Call:
lm(formula = walk ~ program, data = babies)

Residuals:
   Min     1Q Median     3Q    Max 
  -2.4   -0.7   -0.2    0.7    2.8 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)           10.2000     0.5831  17.493 7.46e-12 ***
programRegular         0.4000     0.8246   0.485   0.6342    
programWeekly report   1.2000     0.8246   1.455   0.1649    
programSingle report   1.8000     0.8246   2.183   0.0443 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.304 on 16 degrees of freedom
Multiple R-squared:  0.2639,    Adjusted R-squared:  0.1258 
F-statistic: 1.912 on 3 and 16 DF,  p-value: 0.1684

The first level of the factor is the benchmark.

Applying the zero sum constraint

options(contrasts = c("contr.sum", "contr.poly"))
m2 <- lm(walk ~ program, data = babies)
summary(m2)

Call:
lm(formula = walk ~ program, data = babies)

Residuals:
   Min     1Q Median     3Q    Max 
  -2.4   -0.7   -0.2    0.7    2.8 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  11.0500     0.2915  37.901   <2e-16 ***
program1     -0.8500     0.5050  -1.683    0.112    
program2     -0.4500     0.5050  -0.891    0.386    
program3      0.3500     0.5050   0.693    0.498    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.304 on 16 degrees of freedom
Multiple R-squared:  0.2639,    Adjusted R-squared:  0.1258 
F-statistic: 1.912 on 3 and 16 DF,  p-value: 0.1684

Study: Battery Lifetime

Battery Lifetime Data

id	material	temp	lifetime
1	1	15	130
2	1	15	155
3	1	70	34
4	1	70	40
5	1	125	20
6	1	125	70
7	1	15	74
8	1	15	180
9	1	70	80
10	1	70	75
11	1	125	82
12	1	125	58
13	2	15	150
14	2	15	188
15	2	70	136
16	2	70	122
17	2	125	25
18	2	125	70
19	2	15	159
20	2	15	126
21	2	70	106
22	2	70	115
23	2	125	58
24	2	125	45
25	3	15	138
26	3	15	110
27	3	70	174
28	3	70	120
29	3	125	96
30	3	125	104
31	3	15	168
32	3	15	160
33	3	70	150
34	3	70	139
35	3	125	82
36	3	125	60

Counts in each treatment group

material	temp	n
1	15	4
1	70	4
1	125	4
2	15	4
2	70	4
2	125	4
3	15	4
3	70	4
3	125	4

EDA