Module #11 Assignment

#Part 1: Additive Model for the Ashina Dataset
> library(ISwR)
> data(ashina)
> ashina$subject <- factor(1:16)
> act <- data.frame(vas = ashina$vas.active, subject = ashina$subject, treat = 1, period = ashina$grp)
> plac <- data.frame(vas = ashina$vas.plac, subject = ashina$subject, treat = 0, period = ashina$grp)
> combined <- rbind(act, plac)
> additive_model <- lm(vas ~ subject + treat + period, data = combined)
> summary(additive_model)

Call:
lm(formula = vas ~ subject + treat + period, data = combined)

Residuals:
   Min     1Q Median     3Q    Max 
-48.94 -18.44   0.00  18.44  48.94 

Coefficients: (1 not defined because of singularities)
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -113.06      27.39  -4.128 0.000895 ***
subject2       51.50      37.58   1.370 0.190721    
subject3      121.50      37.58   3.233 0.005573 ** 
subject4       97.00      37.58   2.581 0.020867 *  
subject5      125.00      37.58   3.326 0.004604 ** 
subject6       31.50      37.58   0.838 0.415070    
subject7      119.50      37.58   3.180 0.006215 ** 
subject8      132.00      37.58   3.513 0.003142 ** 
subject9       80.50      37.58   2.142 0.049003 *  
subject10     116.00      37.58   3.087 0.007518 ** 
subject11     121.50      37.58   3.233 0.005573 ** 
subject12     154.50      37.58   4.111 0.000925 ***
subject13     131.00      37.58   3.486 0.003318 ** 
subject14     125.00      37.58   3.326 0.004604 ** 
subject15      99.00      37.58   2.634 0.018768 *  
subject16      80.50      37.58   2.142 0.049003 *  
treat         -42.87      13.29  -3.227 0.005644 ** 
period            NA         NA      NA       NA    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 37.58 on 15 degrees of freedom
Multiple R-squared:  0.7566,	Adjusted R-squared:  0.4969 
F-statistic: 2.914 on 16 and 15 DF,  p-value: 0.02229

> t_test_result <- t.test(ashina$vas.active, ashina$vas.plac, paired = TRUE)

#Part 2: Model Matrices and Interactions
> a <- gl(2, 2, 8)   # Two levels, each repeated twice, for 8 values
> b <- gl(2, 4, 8)   # Two levels, each repeated four times, for 8 values
> x <- 1:8
> y <- c(1:4, 8:5)
> z <- rnorm(8)
> model_matrix_ab <- model.matrix(~ a * b)      # Full interaction model
> model_matrix_ab_interaction <- model.matrix(~ a:b)  # Only interaction terms
> model_matrix_main <- model.matrix(~ a + b)    # Main effects only
> lm_ab <- lm(z ~ a * b)
> lm_ab_interaction <- lm(z ~ a:b)
> lm_main <- lm(z ~ a + b)
> summary(lm_ab)

Call:
lm(formula = z ~ a * b)

Residuals:
      1       2       3       4       5       6       7       8 
-0.6626  0.6626 -0.1294  0.1294 -1.0456  1.0456  0.9444 -0.9444 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.0493     0.7812  -0.063    0.953
a2           -0.5231     1.1048  -0.473    0.661
b2           -0.3548     1.1048  -0.321    0.764
a2:b2         1.7508     1.5624   1.121    0.325

Residual standard error: 1.105 on 4 degrees of freedom
Multiple R-squared:  0.3224,	Adjusted R-squared:  -0.1858 
F-statistic: 0.6344 on 3 and 4 DF,  p-value: 0.6309

> summary(lm_ab_interaction)

Call:
lm(formula = z ~ a:b)

Residuals:
      1       2       3       4       5       6       7       8 
-0.6626  0.6626 -0.1294  0.1294 -1.0456  1.0456  0.9444 -0.9444 

Coefficients: (1 not defined because of singularities)
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.8236     0.7812   1.054    0.351
a1:b1        -0.8729     1.1048  -0.790    0.474
a2:b1        -1.3960     1.1048  -1.264    0.275
a1:b2        -1.2277     1.1048  -1.111    0.329
a2:b2             NA         NA      NA       NA

Residual standard error: 1.105 on 4 degrees of freedom
Multiple R-squared:  0.3224,	Adjusted R-squared:  -0.1858 
F-statistic: 0.6344 on 3 and 4 DF,  p-value: 0.6309

> summary(lm_main)

Call:
lm(formula = z ~ a + b)

Residuals:
      1       2       3       4       5       6       7       8 
-0.2249  1.1003 -0.5671 -0.3083 -1.4833  0.6079  1.3821 -0.5067 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.4870     0.6936  -0.702    0.514
a2            0.3523     0.8009   0.440    0.678
b2            0.5206     0.8009   0.650    0.544

Residual standard error: 1.133 on 5 degrees of freedom
Multiple R-squared:  0.1097,	Adjusted R-squared:  -0.2464 
F-statistic: 0.308 on 2 and 5 DF,  p-value: 0.7479

Interpretation of data:
Full interaction (z ~ a * b): This includes both main effects and the interaction effect, which lead to singularities if a and b are perfectly correlated.
Only interaction (z ~ a:b): Excludes main effects, leading to a simpler model, but may miss main effect contributions.
Main effects only (z ~ a + b): Ignores interactions, assuming a and b contribute independently.