Statistics in medicine One group is measured one time: Lecture 3: - PDF document

11/2/2016 Proportion in one group Statistics in medicine • One group is measured one time: Lecture 3: Bivariate association : Categorical variables – z test • Use the z distribution as an approximation to the binomial distribution Fatma Shebl, MD, MS, MPH, PhD Assistant Professor Chronic Disease Epidemiology Department Yale School of Public Health Fatma.shebl@yale.edu S L I D E 0 S L I D E 1 Steps of the statistical significance Proportion in one group, z test testing, z test • 1- Calculate the test statistic 1- Calculate the critical value • 2- Determine the critical value of • Critical ratio (critical value) significance 𝑞−𝜌  Z = 𝜌(1−𝜌)/𝑜 • 3-Compare the test statistic with the critical value • 4- Calculate the p value • 5- Draw a conclusion S L I D E 2 S L I D E 3 Proportion in one group, z test Proportion in one group, z test 2-Determine the 3-Compare the test statistic with the critical critical value of significance value Critical value: – The test statistic is < the critical value (in – E.g., the critical absolute sense)  acceptance area value for .05 alpha level, one-tailed – The test statistic is > the critical value (in test, is 1.645 absolute sense)  rejection area Source of table: http://www.had2know.com/academics/normal-distribution-table-z-scores.html S L I D E 4 S L I D E 5 1

11/2/2016 Proportion in one group, z test Proportion in one group, z test 4- Calculate the p value 4- Calculate the p value • Using the z-table – E.g., for a z=3.1 – Find the z value that is =test statistic by scrolling down the • one-tailed test, p=(1-.999) z column then scrolling to the right in the row =.001 • two-tailed test, p=(1-.999) – The entries of the tables are used to calculate the p values *2=.002 • For negative z values • One-tailed, the p value is the entry of the z table • Two-tailed, the p value is the entry of the z table multiplied by 2 • For positive z values • One-tailed, the p value = 1- the entry of the z table • Two-tailed,: p value = (1- the entry of the z table )multiplied by 2 Source of table: http://www.had2know.com/academics/normal-distribution-table-z-scores.html S L I D E 6 S L I D E 7 Proportion in one group, z test Proportion in one group, z test 5- Calculate the confidence interval 6- Draw a conclusion • Reject or fail to reject the null hypothesis • CI of proportion: – Fail to reject the null 𝑞(1−𝑞) • If the p value > alpha level – 𝑞 ± 𝑨 × 𝑜 • If the confidence interval crosses the 1 • E.g. – Reject the null • If the p value < alpha level 𝑞(1−𝑞) – 95% CI= 𝑞 ± 1.96 × • If the confidence interval does not cross 𝑜 the 1 The value that defines the central 95% of the area S L I D E 8 S L I D E 9 Proportion in one group, Proportion in one group, z test paired/matched example • A study of success of • One group is measured twice (or matched vaccination revealed that design): 97.1% responded to the vaccine. The investigators – McNemar Chi-square would like to examine whether this proportion is significantly higher than • Is a chi-square test for comparing 95%. If the calculated z value proportions from two DEPENDENT or was 1.75. Using a .05 alpha level, what is the critical value paired groups. and the corresponding p • Use the chi-square distribution value? • Answer: – Critical value=1.645 (one tail) – p=(1-.9599)=.0401 – Conclusion: we reject the Source: http://www.had2know.com/academics/normal-distribution-table-z-scores.html null hypothesis S L I D E 10 S L I D E 11 2

11/2/2016 Proportion in one group, Steps of the statistical significance paired/matched testing, McNemar Chi-square • 1- Calculate the test statistic (critical • One group is measured twice (or matched ratio) design): • 2- Calculate degrees of freedom – McNemar Chi-square • 3- Determine the critical value of • E.g., E.g., Is there a change in bowel significance functions after cholecystectomy? • 4-Compare the test statistic with the critical value • 5- Calculate the p value • 6- Calculate the confidence interval • 7- Draw a conclusion S L I D E 12 S L I D E 13 Proportion in one group, Proportion in one group, paired/matched, McNemar test paired/matched, McNemar test 3-Determine the critical value of 1- Calculate the df Probability (p value) significance 0.10 0.05 0.025 0.01 0.005 critical value Matched design Critical value: 1 2.706 3.841 5.024 6.635 7.879 Cases Control Total • Critical ratio – Depends on: 2 4.605 5.991 7.378 9.210 10.597 Positive risk Negative risk (critical value) • Alpha level 3 6.251 7.815 9.348 11.345 12.838 Positive risk a b a+b • Degrees of freedom Negative risk c d c+d 4 7.779 9.488 11.143 13.277 14.860  McNemar X 2 = Total a+c b+d a+b+c+d – E.g., for an alpha of .05, 1 df, 5 9.236 11.070 12.833 15.086 16.750 2 𝑐−𝑑 the value of the critical x 2 is 𝑐+𝑑 6 10.645 12.592 14.449 16.812 18.548 3.841 7 12.017 14.067 16.013 18.475 20.278 2- Calculate degrees Paired design 8 13.362 15.507 17.535 20.090 21.955 Before intervention After intervention Total of freedom 9 14.684 16.919 19.023 21.666 23.589 Positive Negative 10 15.987 18.307 20.483 23.209 25.188 Positive a b a+b • df= (R-1)(C-1)=1 11 17.275 19.675 21.920 24.725 26.757 Negative c d c+d Total a+c b+d a+b+c+d 12 18.549 21.026 23.337 26.217 28.300 13 19.812 22.362 24.736 27.688 29.819 S L I D E 14 S L I D E 15 Proportion in one group, Proportion in one group, paired/matched, McNemar test paired/matched, McNemar test 5- Calculate the p value df Probability (p value) 4-Compare the test statistic with the critical 0.10 0.05 0.025 0.01 0.005 • From the x 2 table value 1 2.706 3.841 5.024 6.635 7.879 – Locate the row of the 2 4.605 5.991 7.378 9.210 10.597 – The test statistic is < the critical value (in relevant df 3 6.251 7.815 9.348 11.345 12.838 absolute sense)  acceptance area – Find the x 2 value that is 4 7.779 9.488 11.143 13.277 14.860 < test statistic 5 9.236 11.070 12.833 15.086 16.750 – The test statistic is > the critical value (in 6 10.645 12.592 14.449 16.812 18.548 – Find the p value at the absolute sense)  rejection area 7 12.017 14.067 16.013 18.475 20.278 top of the column 8 13.362 15.507 17.535 20.090 21.955 – E.g., for a, 1 df, x 2 =4 9 14.684 16.919 19.023 21.666 23.589 –  .05 > p value > .025 10 15.987 18.307 20.483 23.209 25.188 11 17.275 19.675 21.920 24.725 26.757 12 18.549 21.026 23.337 26.217 28.300 13 19.812 22.362 24.736 27.688 29.819 S L I D E 16 S L I D E 17 3

11/2/2016 Proportion in one group, Proportion in one group, paired/matched, paired/matched, McNemar test McNemar test example Researcher examined df Probability (p value) 6- Draw a conclusion whether bowel movement 0.10 0.05 0.025 0.01 0.005 • Reject or fail to reject the null hypothesis changed postoperative. If the 1 2.706 3.841 5.024 6.635 7.879 calculated McNemar statistic 2 4.605 5.991 7.378 9.210 10.597 – Fail to reject the null is 15, what is: the df and the 3 6.251 7.815 9.348 11.345 12.838 • If the p value > alpha level p value? 4 7.779 9.488 11.143 13.277 14.860 5 9.236 11.070 12.833 15.086 16.750 • df=1 – Reject the null 6 10.645 12.592 14.449 16.812 18.548 • 0.005 > p value • If the p value < alpha level 7 12.017 14.067 16.013 18.475 20.278 8 13.362 15.507 17.535 20.090 21.955 9 14.684 16.919 19.023 21.666 23.589 10 15.987 18.307 20.483 23.209 25.188 11 17.275 19.675 21.920 24.725 26.757 12 18.549 21.026 23.337 26.217 28.300 13 19.812 22.362 24.736 27.688 29.819 S L I D E 18 S L I D E 19 Proportions in two groups, z Readings and resources approximation test • Z test: • Chapter 5, p93-133: Dawson, B. and – Definition: Test of the equality of two Trapp, R. G. (2004). Basic and Clinical independent proportions Biostatistics (4th edition). New York: McGraw-Hill. – Is an approximation of the binomial distribution when p*n>5 • Chapter 11, p134-146: Jekel's epidemiology, biostatistics, preventive – Requires at least 5 observed frequencies in medicine, and public health by David L. each group Katz et al (4th edition). – Parametric test S L I D E 20 S L I D E 21 Steps of the statistical significance Proportions in two groups, z test testing, z test 1- Calculate the critical value • 1- Calculate the test statistic 𝑞 1 −𝑞 2 critical value Z = 𝑞 1−𝑞 [ 1/𝑜 1 + 1/𝑜 2 ] • 2- Determine the critical value of significance Where, • 3-Compare the test statistic with the • 𝑞 1 , 𝑏𝑜𝑒 𝑞 2 are proportions in each group critical value • P is the pooled proportion • 4- Calculate the p value 𝑜 1 𝑞 1 +𝑜 2 𝑞 2 • 5- Draw a conclusion • p = 𝑜 1 +𝑜 2 S L I D E 22 S L I D E 23 4