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Course Information
Meet the Teaching Team -
Course Dataset 1
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Course Dataset 2
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MODULE A1: INTRODUCTION TO STATISTICS USING R, STATA, AND SPSSA1.1 What is Statistics?
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A1.2.1a Introduction to Stata
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A1.2.2b: Introduction to R
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A1.2.2c: Introduction to SPSS
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A1.3: Descriptive Statistics
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A1.4: Estimates and Confidence Intervals
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A1.5: Hypothesis Testing
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A1.6: Transforming Variables
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End of Module A11 Quiz
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MODULE A2: POWER & SAMPLE SIZE CALCULATIONSA2.1 Key Concepts
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A2.2 Power calculations for a difference in means
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A2.3 Power Calculations for a difference in proportions
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A2.4 Sample Size Calculation for RCTs
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A2.5 Sample size calculations for cross-sectional studies (or surveys)
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A2.6 Sample size calculations for case-control studies
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End of Module A21 Quiz
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MODULE B1: LINEAR REGRESSIONB1.1 Correlation and Scatterplots
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B1.2 Differences Between Means (ANOVA 1)
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B1.3 Univariable Linear Regression
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B1.4 Multivariable Linear Regression
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B1.5 Model Selection and F-Tests
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B1.6 Regression Diagnostics
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End of Module B11 Quiz
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MODULE B2: MULTIPLE COMPARISONS & REPEATED MEASURESB2.1 ANOVA Revisited – Post-Hoc Testing
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B2.2 Correcting For Multiple Comparisons
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B2.3 Two-way ANOVA
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B2.4 Repeated Measures and the Paired T-Test
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B2.5 Repeated Measures ANOVA
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End of Module B21 Quiz
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MODULE B3: NON-PARAMETRIC MEASURESB3.1 The Parametric Assumptions
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B3.2 Mann-Whitney U Test
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B3.3 Kruskal-Wallis Test
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B3.4 Wilcoxon Signed Rank Test
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B3.5 Friedman Test
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B3.6 Spearman’s Rank Order Correlation
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End of Module B31 Quiz
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MODULE C1: BINARY OUTCOME DATA & LOGISTIC REGRESSIONC1.1 Introduction to Prevalence, Risk, Odds and Rates
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C1.2 The Chi-Square Test and the Test For Trend
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C1.3 Univariable Logistic Regression
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C1.4 Multivariable Logistic Regression
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End of Module C11 Quiz
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MODULE C2: SURVIVAL DATAC2.1 Introduction to Survival Data
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C2.2 Kaplan-Meier Survival Function & the Log Rank Test
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C2.3 Cox Proportional Hazards Regression
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C2.4 Poisson Regression
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End of Module C21 Quiz
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A Note about the Fossa Certificate
Learning Outcomes
By the end of this section, students will be able to:
- Explain the key concept of power and what impacts it
- Estimate the power of a given study
- Estimate the sample size needed to test hypotheses in different study designs
You can download a copy of the slides here: A2.2 Power calculations for a difference in means
Video A2.2 Power Calculation for a Difference in Means (8 minutes)
A2.2 PRACTICAL: R
Power calculations for two means
Here is an example:
Estimate the sample size needed to compare the mean systolic blood pressure (SBP) in two populations. From a pilot study, you think that the group with lower blood pressure will have a mean SBP of 120 mm Hg, and the standard deviation (SD) of both groups will be 15 mm Hg. You have decided that you are interested in a minimum difference of 5 mm Hg, and you want 90% power, and a 5% significance level.
In R, we need to calculate two statistics to estimate sample size: delta (i.e. the expected difference between groups) and sigma (variance, which in this case is the pooled standard deviation). Once we have delta and sigma, we can calculate the effect size we expect to see, which is Cohen’s d. We can guess delta and sigma from looking at past studies or by running a pilot study. Cohen’s d is estimated by dividing the delta by the sigma.
d <- 5/15
d
[1] 0.3333333
Then we can use the ‘pwr.t.test’ command (from the power package) to assess the sample size needed to detect this effect size.
### d = Cohen’s d
### power = 0.9
### alpha = 0.05
power1<-pwr.t.test(d=d, power=0.9, sig.level =0.05 )
power1
Two-sample t test power calculation
         n = 190.0991
d = 0.3333333
sig.level = 0.05
power = 0.9
alternative = two.sided
NOTE: n is number in *each* group
You need approximately 190 participants in each group, and 380 participants overall.
If we want to estimate the power of a given sample size, we omit the ‘power’ option, and instead use the ‘n=’ option:
>power2<-pwr.t.test(n=190, d=d, sig.level =0.05 )
> power2
Two-sample t test power calculation
         n = 190
d = 0.3333333
sig.level = 0.05
power = 0.8998509
alternative = two.sided
NOTE: n is number in *each* group
We can see here that recruiting 190 participants in each blood pressure group would enable our study to have 90% power.
Question A2_2: Using the same study outlined above, how much power would we have ended up with in our study if we only managed to recruit 150 participants in each group, but the variance of our study sample was smaller than what we anticipated (so SD=12)?
Answer
We first need to recalculate our Cohen’s d (effect size):
d2 <- 5/12
d2
[1] 0.4166667
power3<-pwr.t.test(n=150, d=d2, sig.level =0.05 )
>power3
Two-sample t test power calculation
         n = 150
d = 0.4166667
sig.level = 0.05
power = 0.9491662
alternative = two.sided
NOTE: n is number in *each* group
We recruited fewer participants, which would decrease our power, but since our variance was lower our power actually increased overall to 95%.
A2.2 PRACTICAL: Stata
Power calculations for two means
Here is an example:
Estimate the sample size needed to compare the mean systolic blood pressure (SBP) in two populations. From a pilot study, you think that the group with lower blood pressure will have a mean SBP of 120 mm Hg, and the standard deviation (SD) of both groups will be 15 mm Hg. You have decided that you are interested in a minimum difference of 5 mm Hg, and you want 90% power, and a 5% significance level.
The command and output is as follows:
power twomeans 120, power(0.9) alpha(0.05) diff(5) sd(15)
*– Estimated sample sizes:
           N =      382
 N per group =      191
*– Estimated sample size: 382 (191 per group).
You need approximately 382 participants overall.
If we want to estimate the power of a given sample size, we omit the ‘power’ option, and instead use the ‘n( )’ option:
power twomeans 120, alpha(0.05) diff(5) sd(15) n(382)
Question A2.2: Using the same study outlined above, how much power would we have ended up with in our study if we only managed to recruit 300 participants in total, but the variance of our study sample was smaller than what we anticipated (so SD=12)?
Answer
power twomeans 120, alpha(0.05) diff(5) sd(12) n(300)
We recruited fewer participants, which would decrease our power, but since our variance was lower our power actually increased overall to 95%.
A2.2 PRACTICAL: SPSS
Power calculations for two means
Here we want to estimate the sample size needed to compare the mean systolic blood pressure (SBP) in two populations. From a pilot study, you think that the group with lower blood pressure will have a mean SBP of 120 mm Hg, and the standard deviation (SD) of both groups will be 15 mm Hg. You have decided that you are interested in a minimum difference of 5 mm Hg, and you want 90% power, and a 5% significance level.
Select
Analyze >> Power Analysis >> Means >> Independent Samples T Test
In the Power Analysis window, you need to enter the following:
- Estimate: Sample size (because this is what we want to calculate)
- Single power value: 0.9 (we are looking for 90% power, and this is shown as a decimal)
- Population mean difference: 5 (the difference we are looking for)
- Population standard deviation: 15 (use the equal for two groups option, as we would not expect the groups to differ from each other.
- Significance level (α): 0.05
If we want to estimate the power of a given sample size, we open the power analysis window in the same way, but select power from the drop-down menu at the top instead of sample size. Input the sample size for each of your two groups at the top of the box, then input all the rest of the values as before. Then press OK to run the test.
Using the same study outlined above, how much power would we have ended up with in our study if we only managed to recruit 300 participants in total, but the variance of our study sample was smaller than what we anticipated (so SD=12)?
Answer
For the first part of the question the output table would look like this.
Estimated sample size is 382 (191 per group), so you need approximately 382 participants overall for the study to have your desired power of 90%
For the second part of the question the output table would like like this.
We recruited fewer participants, which would decrease our power, but since our variance was lower our power actually increased overall to 95%.
Good progress