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

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?)

2 Can we quantify how much power we have for given sample sizes

Can we quantify how much power we have for given sample sizes

3 study 1: 263 cases, 1241 controls

study 1: 263 cases, 1241 controls

Null Distribution: difference=0.

Clinically relevant alternative: difference=10%.

4 study 1: 263 cases, 1241 controls

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)

5 study 1: 50 cases, 50 controls

study 1: 50 cases, 50 controls

Power closer to 15% now.

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

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

Power is nearly 100%!

Clinically relevant alternative: difference=4 points

7 Study 2: 18 treated, 72 controls, STD DEV=10

Study 2: 18 treated, 72 controls, STD DEV=10

Power is about 40%

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

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

Power is about 50%

Clinically relevant alternative: difference=1 point

9 Factors Affecting Power

Factors Affecting Power

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

10 1. Bigger difference from the null mean

1. Bigger difference from the null mean

11 2. Bigger standard deviation

2. Bigger standard deviation

12 3. Bigger Sample Size

3. Bigger Sample Size

13 4. Higher significance level

4. Higher significance level

14 Sample size calculations

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**

15 Simple formula for difference in means

Simple formula for difference in means

16 Simple formula for difference in proportions

Simple formula for difference in proportions

17 Derivation of sample size formula…

Derivation of sample size formula…

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

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

Power close to 50%

19 SAMPLE SIZE AND POWER FORMULAS

SAMPLE SIZE AND POWER FORMULAS

20 Power is the area to the right of Z

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.

21 All-purpose power formula…

All-purpose power formula…

22 Derivation of a sample size formula…

Derivation of a sample size formula…

Sample size is embedded in the standard error….

23 Algebra…

Algebra…

24 Introduction to sample size and power calculations
25 Sample size formula for difference in means

Sample size formula for difference in means

26 Examples

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:

27 Power formula…

Power formula…

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

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

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

29 Sample Size needed for comparing two proportions:

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.

30 Derivation of a sample size formula:

Derivation of a sample size formula:

The standard error of the difference of two proportions is:

31 Derivation of a sample size formula:

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)=

32 Introduction to sample size and power calculations
33 For 80% power…

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.

34 Question 2:

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?

35 Different size groups…

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.

36 General sample size formula

General sample size formula

37 General sample size needs when outcome is binary:

General sample size needs when outcome is binary:

38 Compare with when outcome is continuous:

Compare with when outcome is continuous:

39 Question

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)?

40 Introduction to sample size and power calculations
41 Therefore, need: (9)(1

Therefore, need: (9)(1

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

42 Sample size for paired data:

Sample size for paired data:

43 Paired data difference in proportion: sample size:

Paired data difference in proportion: sample size:

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