Lecture 7 - Repeated measures Flashcards
Name a research example where a repeated measure test can be used with one group and give the associated formula for it.
Investigating the change in cholesterol over time
* mean cholesterol = b0 + b1* time
To investigate the change in cholesterol over time two tests can be used. Name and describe these.
- Repeated measures ANOVA, each measurement is put in a separate column for each patient in a row (wide/short format). Correction for confounding is not possible.
- Linear mixed model, each measurement is put in a separate row (so three time measurements per patient is equal to three rows of time measurements for each patient). Correction for confounding is possible.
What assumptions need to be met in order to perform a repeated measures ANOVA?
- Normality of the outcome
- Sphericity: equal variances at different time points and equal correlations between different time points.
How can sphericity as an assumption for a repeated measures ANOVA be tested?
With the use of the Mauchly’s Test of Sphericity, which determines what value to look at in the table of ‘Tests of Within-Subjects Effects’. If the p-value is larger than 0.05, sphericity is assumed. If the p-value is smaller than 0.05, the Greenhouse-Geisser should be used for analysis.
What information can you get from the ‘Tests of Within-Subjects Effects’ table after performing a repeated measure ANOVA with one group?
How much of the variation in the outcome (e.g. cholesterol) is explained by time (partial eta squared).
If you’re investigating the change in cholesterol over time, measured by 6 different time points, you can calculate how much of the variation in cholesterol levels is explained by time. In addition, you could also research at which time points the variation is strongest by doing a post-hoc analysis. This can be done by using the Tests of Within-Subjects Contrasts. Given te following p-values, explain/calculate which time points have the strongest cholesterol changes.
- T1 vs. T2: p-value=0.004
- T2 vs. T3: p-value=0.136
- T3 vs. T4: p-value=0.005
- T4 vs. T5: p-value=0.000
- T5 vs. T6: p-value=0.000
There are in total 5 p-values calculated, so every significantp-value is multiplied by 5.
* T1 vs. T2: p-value=0.004 x 5 = 0.02
* T2 vs. T3: p-value=0.136 -> no significant change in cholesterol
* T3 vs. T4: p-value=0.005 x 5 = 0.025
* T4 vs. T5: p-value=0.000 x 5 = 0.000
* T5 vs. T6: p-value=0.000 x 5 = 0.000
Therefore, the strongest change in cholesterol is T4 vs. T5 and T5 vs. T6.
A repeated measures ANOVA is performed to investigate whether there is a difference in change of cholesterol over time between males and females. Describe:
* formula
* within subjects variable
* between subjects variable
* null hypothesis
* alternative hypothesis
- formula: b0 + b1 x sex + b2 x time + b3 x sex x time
- within subjects variable: time
- between subjects variable: sex
- null hypothesis: b3 is equal to 0
- alternative hypothesis: b3 is not equal to 0
A repeated measures ANOVA is performed to investigate whether there is a difference in change of cholesterol over time between males and females.
To determine whether the null hypothesis is true or not, the ‘Test of Within-Subjects Effects’ table is used. By looking at the second row (T × sex), you can conclude that there is a significant difference in change of cholesterol over time between males and females.
What cannot be concluded based on this table?
Whether cholesterol differs between males and females at e.g. T1 vs. T2. For this, you have to use a paired sample t-test with correction for multple testing (post-hoc analysis). Note that this is not an option in SPSS for comparing difference between 2 (or more groups) at different time points.
Besides a repeated measures ANOVA for analysing the change in cholesterol over time, a linear mixed model can also be used for this. Describe the formula for the following research question: “Does cholesterol change over time?”
Mean cholesterol = b0 + b1 x time
A (linear) mixed model uses random and fixed variables. Explain this.
If it is researched whether cholesterol changes over time, the dependent variable is the change in cholesterol and the independent variable is time. The ‘problem’ here is that the patients whom cholesterol was measured cannot be added as categorical covariates, becauase there will be too many categories (for example when cholesterol levels have been measured at 10 different time points in 100 patients, meaning there will be at least 1000 cholesterol measurements). For this, the patients are added as a random effect and the fixed effect is time.
In this way:
* fixed effect represents the average change in cholesterol over time across the entire population.
* random effect accounts for the individual variation among patients so that you acknowledge that there are individual differences in baseline cholesterol levels and how they respond to changes over time.
The formula for the research question if cholesterol changes over time is equal to: mean cholesterol = b0 + b1 x time.
Suppose time is treated as a categorical covariate and there are 6 time points, describe the formula for this.
So if there are 6 time points, these have to be recoded into 5 dummies for time:
mean cholesterol = b0 + a2 × dT2 + a3 × dT3 + a4 × dT4 + a5 × dt5 + a6 × dT6
* b0 = mean choleserol at T1
* ai = mean change in cholesterol between Ti and T1
Describe the formula for the following research question: Is there a difference in the change of cholesterol over time between males and females?
Mean cholesterol = b0 + b1 × sex + b2 × time + b3 × sex × time
Describe the fixed and random effect for the following research question:
Is there a difference in the change of cholesterol over time between males and females (Mean cholesterol = b0 + b1 × sex + b2 × time + b3 × sex × time).
- Fixed effect: sex × time
- Random effect: patients
The formula for the research question if there is a difference incholesterol change over time between males and females is equal to: mean cholesterol = b0 + b1 × sex + b2 × time + b3 × sex × time.
Suppose time is treated as a categorical covariate and there are 6 time points, then the formula is equal to: mean cholesterol = b0 + b1 × sex + a2 × dT2 + a3 × dT3 + a4 × dT4 + a5 × dT5 + a6 × dT6 + c2 × sex × dT2 + c3 × sex × dT3 + c4 × sex × dT4 + c5 × sex × dT5 + c6 × sex × dT6
Describe for the following ‘variables’ what they mean:
* b0
* b0 + a6
* b0 + b1
* b0 + b1 + a6 + c6
* b1
* ai
* ci
* ai + ci
Female = 0, male = 1
- b0: mean cholesterol of women at T1
- b0 + a6 = mean cholesterol of women at T6
- b0 + b1 = mean cholesterol of men at T1
- b0 + b1 + a6 + c6 = mean cholesterol of men at T6
- b1 = mean difference between men and women at T1
- ai = mean change within women between Ti and T1
- ci = mean difference in change between men and women between Ti and T1
- ai + ci = mean change within men between Ti and T1
Describe the formula for the following research question: Is there a difference in the change of cholesterol over time between males and females, after correcting for age?
Mean cholesterol = b0 + b1 × sex + a × time + c × sex × time + b4 × age
