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# Hypothesis Testing and Sample Size Calculation Po Chyou, Ph. D. Director, BBC.

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Hypothesis Testing and Sample Size Calculation Po Chyou, Ph. D. Director, BBC

Hypothesis Testing on Population mean(s) Population median(s)  Population proportion(s) Population variance(s) Population correlation(s)  Association based on contingency table(s) Coefficients based on regression model Odds ratio Relative risk Trend analysis Survival distribution(s) / curve(s) Goodness of fit

Hypothesis Testing 1.Definition of a Hypothesis An assumption made for the sake of argument 2.Establishing Hypothesis Null hypothesis - H 0 Alternative hypothesis - H a 3. Testing Hypotheses Is H 0 true or not?

Hypothesis Testing 4.Type I and Type II Errors Type I error: we reject H 0 but H 0 is true α = Pr(reject H 0 / H 0 is true) = Pr(Type I error) = Level of significance in hypothesis testing Type II error: we accept H 0 but H 0 is false  = Pr(accept H 0 / H 0 is false) = Pr(Type II error)

Hypothesis Testing 5.Steps of Hypothesis Testing - Step 1Formulate the null hypothesis H 0 in statistical terms - Step 2Formulate the alternative hypothesis H a in statistical terms - Step 3Set the level of significance α and the sample size n - Step 4Select the appropriate statistic and the rejection region R - Step 5Collect the data and calculate the statistic

Hypothesis Testing 5.Steps of Hypothesis Testing (continued) - Step 6If the calculated statistic falls in the rejection region R, reject H 0 in favor of H a ; if the calculated statistic falls outside R, do not reject H 0

Hypothesis Testing A random sample of 400 persons included 240 smokers and 160 non- smokers. Of the smokers, 192 had CHD, while only 32 non-smokers had CHD. Could a health insurance company claim the proportion of smokers having CHD differs from the proportion of non-smokers having CHD? 6. An Example

Hypothesis Testing Example (continued) Let P 1 = the true proportion of smokers having CHD P 2 = the true proportion of non-smokers having CHD - Step 1 H 0 : P 1 = P 2 - Step 2 H a : P 1  P 2 - Step 3 α =.05, n = 400

Hypothesis Testing Example (continued) - Step 4statistic =  = P 1 - P 2 where P 1 = x 1, P 2 = x 2 and P = x 1 + x 2 n 1 n 2 n 1 + n 2 P(1-P) (1/n 1 + 1/n 2 )

Hypothesis Testing Example (continued) P 1 = x 1 n1n1 = 192 =.80 240 P 2 = x 2 n2 n2 = 32 =.20 160 P = x 1 + x 2 n 1 + n 2 = 192 + 32 = 224 = 0.56 240 + 160 400  = P 1 - P 2 P(1-P) (1/n 1 + 1/n 2 ) =.80 -.20 =.60 = 11.84 > 1.96 (.56) (1-.56) (1/240 + 1/160).05066 - Step 5

Hypothesis Testing Example (continued) - Step 6 Reject H 0 and conclude that smokers had significantly higher proportion of CHD than that of non-smokers. [P-value <.0000001]

Hypothesis Testing 7. Contingency Table Analysis The Chi-square distribution (  2 )

Hypothesis Testing Equation for chi-square for a contingency table For i = 1, 2 and j =1, 2  2 = (O 11 - E 11 ) 2 + (O 12 - E 12 ) 2 + (O 21 - E 21 ) 2 + (O 22 - E 22 ) 2 E 11 E 12 E 21 E 22  2 =  (O ij - E ij ) 2 i, j E ij

Hypothesis Testing Equation for chi-square for a contingency table (cont.) E 11 = n 1 m 1 E 12 = n 1 - n 1 m 1 = n 1 m 2 nnn E 21 = n 2 m 1 E 22 = n 2 - n 2 m 1 = n 2 m 2 nnn

Hypothesis Testing Example : Same as before - Step 1 H 0 : there is no association between smoker status and CHD - Step 2 H a : there is an association between smoker status and CHD - Step 3  =.05, n = 400 - Step 4statistic =  2 = (O 11 - E 11 ) 2 + (O 12 - E 12 ) 2 + (O 21 - E 21 ) 2 + (O 22 - E 22 ) 2 E 11 E 12 E 21 E 22

Hypothesis Testing Example (continued) : Same as before

- Step 5

Hypothesis Testing Example (continued) : Same as before E 11 = n 1 m 1 = 240 * 224 = 134.4 n 400 E 12 = n 1 - n 1 m 1 = 240 - 134.4 = 105.6 n E 21 = n 2 m 1 = 160 * 224 = 89.6 n 400 E 22 = n 2 - n 2 m 1 = 160 - 89.6 = 70.4 n - Step 5 (continued) Expectation Counts

Hypothesis Testing Example (continued) : Same as before - Step 5 (continued)  2 = (O 11 - E 11 ) 2 + (O 12 - E 12 ) 2 + (O 21 - E 21 ) 2 + (O 22 - E 22 ) 2 E 11 E 12 E 21 E 22 = (192 - 134.4) 2 + (48 - 105.6) 2 + (32 - 89.6) 2 + (128 - 70.4) 2 134.4 105.6 89.6 70.4 = 24.68 + 31.42 + 37.03 + 47.13 = 140.26 > 3.841

Hypothesis Testing Example (continued) : Same as before - Step 6 Reject H 0 and conclude that there is an association between smoker status and CHD. [P-value <.0000001]

Sample Size Estimation and Statistical Power Calculation Definition of Power Recall :  = Pr (accept H 0 / H 0 is false) = Pr (Type II error) Power = 1 -  = Pr(reject H 0 / H 0 is false)

Sample Size Estimation for Intervention on Tick Bites Among Campers Assumptions 1.Given that the proportion (P CON ) of tick bites among campers in the control group is constant. 2.Given that the proportion (P INT ) of tick bites among campers in the intervention group is reduced by 50% compared to that of the control group after intervention has been implemented. 3.Given that a one- or two- tailed test is of interest with 80% power and a type-I error of 5%.

Sample Size Estimation for Intervention on Tick Bites Among Campers Summary Table 1

Statistical Power Calculation for Intervention on Obesity of Women in MESA Assumptions 1.Given that the proportion (P CON ) of women who are obese at baseline (i.e., the control group) is constant. There are a total of 840 women in the control group. Based on our preliminary data analysis results, approximately 50% of these 840 women at baseline are obese (BMI >= 27.3). 2.Given that the proportion (P INT ) of women who are obese in the intervention group is reduced by 5% or more compared to that of the control group after intervention has been implemented. There are a total of 680 women who had been newly recruited. Based on our preliminary data analysis results, 50% of these 680 newly recruited women are obese. Assume that 60% of these women will agree to participate, we will have 200 women to be targeted for intervention.

Statistical Power Calculation for Intervention on Obesity of Women in MESA (continued) 3.Given that a one-tailed test is of interest with a type-I error of 5%, then the estimated statistical powers are shown in Table 1 for detecting a difference of 5% or more in the proportion of obesity between the control group and the intervention group. Assumptions Table 1

Reference “Statistical Power Analysis for the Behavioral Sciences” Jacob Cohen Academic Press, 1977

Take Home Message: You’ve got questions : Data ? S T A TI S T I C S ?... Contact Biostatistics and consult with an experienced biostatistician –Po Chyou, Director, Senior Biostatistician (ext. 9-4776) –Dixie Schroeder, Secretary (ext. 1-7266) OR Do it at your own risk

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