Topic 9B: Repeated measures analysis


These are the solutions for the Topic 9B Computer Lab.

1 Repeated Measures ANOVA

1.1

No solution required.

1.2

No solution required.

1.3

No solution required (see video provided in question sheet to compare your answers).

1.4

  1. \(H_0 : \mu_1 = \mu_2 = \mu_3 \text{ versus } H_1 : \text{not all } \mu_i \text{'s are equal, where}\)

    • \(\mu_1, \mu_2, \mu_3\) denote the population mean self-esteem score for time-point 1, time-point 2, and time-point 3 respectively.

  2. The dependent (response) variable is self esteem score

  3. The independent variable is time

1.5

Consider the following results, which include some R output:

## 
## Error: id
##           Df Sum Sq Mean Sq F value Pr(>F)
## Residuals  9   4.57  0.5078               
## 
## Error: Within
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## time       2 102.46   51.23   55.47 2.01e-08 ***
## Residuals 18  16.62    0.92                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##  Pairwise comparisons using paired t tests 
## 
## data:  selfesteem.long$score and selfesteem.long$time 
## 
##    t1     t2    
## t2 0.0023 -     
## t3 1e-06  0.0027
## 
## P value adjustment method: bonferroni

1.6

  1. Yes. Since we have we have \(p < .001\), we enough evidence to conclude that there is a statistically significant difference in self-esteem scores across time.
  2. \(d_1 = r - 1 = 3 - 1 = 2\). We could read from the ANOVA output, from the Df column, time row.
  3. \(d_2 = (n - 1)(r - 1) = (10 - 1)(3 - 1) = 9\times 2 = 18\). We could read from the ANOVA output, from the Df column, Residuals row.
  4. There was a significant difference in mean self-esteem score [\(F(2, 18) = 55.47\), \(p < .001\)] across time.

1.7

Based on the output, there was a statistically significant difference in mean self-esteem scores between all pairs of time-points. In particular:

  • There was a statistically significant difference between time-point 1 and time-point 2 (\(p = 0.002\))
  • There was a statistically significant difference between time-point 1 and time-point 3 (\(p < .001\))
  • There was a statistically significant difference between time-point 2 and time-point 3 (\(p = 0.003\))

References


These notes have been prepared by Amanda Shaker. The copyright for the material in these notes resides with the authors named above, with the Department of Mathematical and Physical Sciences and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License BY-NC-ND.