Module # 11 assignment
Issaiah Jennings
Module # 11 assignment
> # Load the necessary package and data
> library(ISwR)
> data(ashina)
>
> # Set up factors and prepare data
> ashina$subject <- factor(1:16)
> attach(ashina)
> act <- data.frame(vas = vas.active, subject = subject, treat = 1, period = grp)
> plac <- data.frame(vas = vas.plac, subject = subject, treat = 0, period = grp)
> combined_data <- rbind(act, plac)
>
> # Fit the additive model and view summary
> additive_model <- lm(vas ~ subject + treat + period, data = combined_data)
> summary(additive_model)
Call:
lm(formula = vas ~ subject + treat + period, data = combined_data)
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
>
> # Perform paired t-test for comparison
> t.test(vas.active, vas.plac, paired = TRUE)
Paired t-test
data: vas.active and vas.plac
t = -3.2269, df = 15, p-value = 0.005644
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-71.1946 -14.5554
sample estimates:
mean difference
-42.875
> # Define factors a and b
> a <- gl(2, 2, 8) # Creates factor levels (1, 1, 2, 2) repeated
> b <- gl(2, 4, 8) # Creates factor levels (1, 1, 1, 1, 2, 2, 2, 2)
>
> # Define other variables
> x <- 1:8
> y <- c(1:4, 8:5)
> z <- rnorm(8)
>
> # Generate model matrices for interaction terms
> matrix_a_b <- model.matrix(~ a * b)
> matrix_a_colon_b <- model.matrix(~ a:b)
>
> # Fit linear models with interaction terms
> model_a_b <- lm(z ~ a * b)
> model_a_colon_b <- lm(z ~ a:b)
>
> # Display summaries of the models
> summary(model_a_b)
Call:
lm(formula = z ~ a * b)
Residuals:
1 2 3 4 5 6 7 8
0.54124 -0.54124 0.22015 -0.22015 -0.28026 0.28026 0.01003 -0.01003
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.8493 0.3241 -2.621 0.0587.
a2 0.9411 0.4583 2.054 0.1093
b2 1.6872 0.4583 3.682 0.0212 *
a2:b2 -1.2399 0.6481 -1.913 0.1283
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4583 on 4 degrees of freedom
Multiple R-squared: 0.7948, Adjusted R-squared: 0.6408
F-statistic: 5.163 on 3 and 4 DF, p-value: 0.07335
> summary(model_a_colon_b)
Call:
lm(formula = z ~ a:b)
Residuals:
1 2 3 4 5 6 7 8
0.54124 -0.54124 0.22015 -0.22015 -0.28026 0.28026 0.01003 -0.01003
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.5391 0.3241 1.664 0.1715
a1:b1 -1.3884 0.4583 -3.030 0.0388 *
a2:b1 -0.4473 0.4583 -0.976 0.3843
a1:b2 0.2988 0.4583 0.652 0.5500
a2:b2 NA NA NA NA
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4583 on 4 degrees of freedom
Multiple R-squared: 0.7948, Adjusted R-squared: 0.6408
F-statistic: 5.163 on 3 and 4 DF, p-value: 0.07335
Model Summary
- Min/Max Residuals: Lowest and highest values of residuals, reflecting model fit accuracy.
- 1Q/3Q: First and third quartiles of residuals, marking lower and upper ranges.
- Median: Middle residual value.
- Estimate: Coefficient for each predictor, showing its influence on the outcome.
- Std. Error: Variability in each coefficient’s estimate.
- t Value: Tests if each coefficient significantly differs from zero.
- Pr(>|t|): P-value for the t-test, indicating statistical significance.
- Residual Standard Error**: Average distance of observed values from the regression line.
- Multiple R-squared: Proportion of outcome variance explained by the model.
- Adjusted R-squared: R-squared adjusted for predictor count and sample size.
- F-statistic and p-value: Assess the model’s overall statistical significance.
---
Comparing Models: `z ~ a * b` and `z ~ a:b`
- `z ~ a * b`: Includes main effects of `a` and `b` and their interaction, showing both individual and combined effects on `z`.
- Equation: `z = Intercept + (Effect of a) + (Effect of b) + (a:b Interaction) + Error`
- `z ~ a:b`: Only the interaction term (`a:b`), excluding individual effects, meaning `z` is modeled solely on the joint influence of `a` and `b`.
- Equation: `z = Intercept + (a:b Interaction) + Error`
Comparison of `t.test()` vs `lm()`
- `t.test()`: Ideal for simple mean difference testing between two groups; provides a t-value and p-value.
- `lm()`: Versatile, enabling analysis of multiple predictors and interactions. It’s useful for examining complex models beyond simple mean differences, giving insights into each predictor's effect and interactions on the outcome.
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