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 Introduction to sample size and power calculations Can we quantify how much power we have for given sample sizes study 1: 263 cases, 1241 controls study 1: 263 cases, 1241 controls study 1: 50 cases, 50 controls Study 2: 18 treated, 72 controls, STD DEV = 2 Study 2: 18 treated, 72 controls, STD DEV=10 Study 2: 18 treated, 72 controls, effect size=1 Factors Affecting Power 1. Bigger difference from the null mean 2. Bigger standard deviation 3. Bigger Sample Size 4. Higher significance level Sample size calculations Simple formula for difference in means Simple formula for difference in proportions Derivation of sample size formula… Study 2: 18 treated, 72 controls, effect size=1 SAMPLE SIZE AND POWER FORMULAS Power is the area to the right of Z All-purpose power formula… Derivation of a sample size formula… Algebra… Introduction to sample size and power calculations Sample size formula for difference in means Examples Power formula… Example 2: How many people would you need to sample in each group to Sample Size needed for comparing two proportions: Derivation of a sample size formula: Derivation of a sample size formula: Introduction to sample size and power calculations For 80% power… Question 2: Different size groups… General sample size formula General sample size needs when outcome is binary: Compare with when outcome is continuous: Question Introduction to sample size and power calculations Therefore, need: (9)(1 Sample size for paired data: Paired data difference in proportion: sample size:

Презентация: «Introduction to sample size and power calculations». Автор: kristinc. Файл: «Introduction to sample size and power calculations.ppt». Размер zip-архива: 257 КБ.

## Introduction to sample size and power calculations

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### Introduction to sample size and power calculations

How much chance do we have to reject the null hypothesis when the alternative is in fact true? (what’s the probability of detecting a real effect?)

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### study 1: 263 cases, 1241 controls

Null Distribution: difference=0.

Clinically relevant alternative: difference=10%.

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### study 1: 263 cases, 1241 controls

Power= chance of being in the rejection region if the alternative is true=area to the right of this line (in yellow)

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### study 1: 50 cases, 50 controls

Power closer to 15% now.

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### Study 2: 18 treated, 72 controls, STD DEV = 2

Power is nearly 100%!

Clinically relevant alternative: difference=4 points

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### Study 2: 18 treated, 72 controls, effect size=1

Clinically relevant alternative: difference=1 point

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### Factors Affecting Power

1. Size of the effect 2. Standard deviation of the characteristic 3. Bigger sample size 4. Significance level desired

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### Sample size calculations

Based on these elements, you can write a formal mathematical equation that relates power, sample size, effect size, standard deviation, and significance level… **WE WILL DERIVE THESE FORMULAS FORMALLY SHORTLY**

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### Study 2: 18 treated, 72 controls, effect size=1

Power close to 50%

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### Power is the area to the right of Z

. OR power is the area to the left of - Z?. Since normal charts give us the area to the left by convention, we need to use - Z? to get the correct value. Most textbooks just call this “Z?”; I’ll use the term Zpower to avoid confusion.

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### Derivation of a sample size formula…

Sample size is embedded in the standard error….

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### Examples

= 2.57

Example 1: You want to calculate how much power you will have to see a difference of 3.0 IQ points between two groups: 30 male doctors and 30 female doctors. If you expect the standard deviation to be about 10 on an IQ test for both groups, then the standard error for the difference will be about:

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### Power formula…

P(Z? -.79) =.21; only 21% power to see a difference of 3 IQ points.

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### Example 2: How many people would you need to sample in each group to

achieve power of 80% (corresponds to Z?=.84)

174/group; 348 altogether

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### Sample Size needed for comparing two proportions:

Example: I am going to run a case-control study to determine if pancreatic cancer is linked to drinking coffee. If I want 80% power to detect a 10% difference in the proportion of coffee drinkers among cases vs. controls (if coffee drinking and pancreatic cancer are linked, we would expect that a higher proportion of cases would be coffee drinkers than controls), how many cases and controls should I sample? About half the population drinks coffee.

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### Derivation of a sample size formula:

The standard error of the difference of two proportions is:

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### Derivation of a sample size formula:

Here, if we assume equal sample size and that, under the null hypothesis proportions of coffee drinkers is .5 in both cases and controls, then s.e.(diff)=

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### For 80% power…

Would take 392 cases and 392 controls to have 80% power! Total=784

There is 80% area to the left of a Z-score of .84 on a standard normal curve; therefore, there is 80% area to the right of -.84.

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### Question 2:

How many total cases and controls would I have to sample to get 80% power for the same study, if I sample 2 controls for every case? Ask yourself, what changes here?

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### Different size groups…

Need: 294 cases and 2x294=588 controls. 882 total. Note: you get the best power for the lowest sample size if you keep both groups equal (882 > 784). You would only want to make groups unequal if there was an obvious difference in the cost or ease of collecting data on one group. E.g., cases of pancreatic cancer are rare and take time to find.

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### Question

How many subjects would we need to sample to have 80% power to detect an average increase in MCAT biology score of 1 point, if the average change without instruction (just due to chance) is plus or minus 3 points (=standard deviation of change)?

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### Therefore, need: (9)(1

96+.84)2/1 = 70 people total

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### Paired data difference in proportion: sample size:

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