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Biostatistics course Part 11 Comparison of two proportions. Dr. Sc. Nicolas Padilla Raygoza Department of Nursing and Obstetrics Division of Health Sciences and Engineering Campus Celaya-Salvatierra Universidad de Guanajuato Mexico. Biosketch.
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Biostatistics coursePart 11Comparison of two proportions Dr. Sc. Nicolas Padilla Raygoza Department of Nursing and Obstetrics Division of Health Sciences and Engineering Campus Celaya-Salvatierra Universidad de Guanajuato Mexico
Biosketch • Medical Doctor by University Autonomous of Guadalajara. • Pediatrician by the Mexican Council of Certification on Pediatrics. • Postgraduate Diploma on Epidemiology, London School of Hygiene and Tropical Medicine, University of London. • Master Sciences with aim in Epidemiology, Atlantic International University. • Doctorate Sciences with aim in Epidemiology, Atlantic International University. • Associated Professor B, School of Nursing and Obstetrics of Celaya, university of Guanajuato. • padillawarm@gmail.com
Competencies • The reader will apply a Z test to obtain inferences from two independent proportions. • He (she) will calculate confidence interval from two independent proportions.
Introduction • Often, we make comparisons of two proportions from independent samples. • In class we learned earlier to calculate confidence intervals and hypothesis test for a proportion; we can use the same methods to make inferences on proportions, if the sample size is large. • For a large sample we can use a Normal approximation to the binomial distribution.
Examples • In a study of urinary tract infection not complicated, patients were assigned to be treated with trimethoprim / sulfamethoxazole and fosfomycin / trometamol. • 92 of 100 treated with fosfomycin / trometamol showed bacteriological cure while 61 of 100 treated with trimethoprim / sulfamethoxazole were cured infection.
Introduction • When comparing proportions of independent samples, we must first calculate the difference in proportions. • Analysis to compare two independent proportions is similar to that used for two independent means. • We calculate a confidence interval and hypothesis test for difference in proportions.
Notation • The notation we use for analysis of two proportions is the same as that for a proportion. • The numbers below are for distinguishing the two groups.
Inferences from two independent proportions • The square of the standard error of a proportion is known as the variance of proportion. • The variance of the difference between two independent proportions is equal to the sum of the variances of the proportions of each sample. • The variances are summed because each sample contributes to sampling error in the distribution of differences.
Inferences from two independent proportions • SE = √p(1-p)/n Variance = p(1-p)/n p1(1- p1) p2(1- p2) Variance(p1-p2)= variance of p1 + variance of p2 = --------- + ---------- n1n2 The standard error of the difference between two proportions is given by the square root of the variances. SE(p1-p2)= √[p1(1-p1)/n1 + p2(1-p2)/n2]
Confidence intervals for two independent proportions • To calculate the confidence interval we need to know the standard error of the difference between two proportions. • The standard error of the difference between two proportions is the combination of the standard error of two independent distributions, ES (p1) and (p2). • We estimated the magnitude of the difference of two proportions from the samples; now, calculate the confidence interval for this estimate.
Confidence intervals for two independent proportions • The general formulae for confidence interval 95% is: Estimate ±1.96 x SE • The formulae for IC 95% of two proportions should be: (p1-p2) ± 1.96 SE(p1-p2)
Confidence intervals for two independent proportions • In the study of urinary tract infection, the proportion in the group of fosfomycin / trometamol was 0.92 and trimethoprim / sulfamethoxazole was 0.61 • Difference in proportions = 0.92-0.61 = 0.31 • ES = √ [(0.92 (1-0.92) / 100 + 0.61 (1-0.61) / 100] = 0056 • The confidence interval at 95% would be: • 0.31 ± 1.96 (0,056) = 0.31 ± 0.11 = 0.2 to 0.42
Confidence intervals for two independent proportions • The confidence interval at 95% would be: • 0.31 ± 1.96 (0,056) = 0.31 ± 0.11 = 0.2 to 0.42 • I have 95% confidence that the difference in the proportions in the population would be between 0.2 and 0.42. • As the difference does not include 0, we are confident that the proportion of the population treated with fosfomycin / trometamol is different than with trimethoprim sulfamethoxazole.
Hypothesis test for two independent proportions • A hypothesis test uses the difference and standard error of difference. • However, we use a slightly different standard error to calculate the hypothesis test. • This is because we are assessing the probability that the observed data assume that the null hypothesis is true. • The null hypothesis is that there is no difference in the proportions of both samples and both groups have a common π.
Hypothesis test for two independent proportions • The best estimate we can get from π is the common proportion, p of the two proportions of the sample. P = r1 + n2 + r2/n1+n2 • Where: • r1 and r2 are numbers of positive responses in each sample • n1 and n2 are the sample sizes in each sample. • Common proportion will be between two individual proportions.
Hypothesis test for two independent proportions • The standard error can be calculated by replacing p by p1 and p2. • SE(p1-p2)=√p(1-p)(1/n1 +1/n2) • This is known as a pooled standard error.
Example • In the study of urinary tract infection, the proportion in the group of fosfomycin / trometamol was 0.92 and trimethoprim / sulfamethoxazole was 0.61 • 100 integrants were in each group. • Common p = 92 + 61/100 + 100 = 153/200 = 0.765 • SE (p1-p2) = √ 0.77 (1-0.77) (1 / 100 +1 / 100) = √ 0.1771 x 0.002 = 0.019
Example • Assuming a normal approximation to the binomial distribution, we calculate the Z test, as before. • To calculate the hypothesis test, we must: 1 .- Identify the null hypothesis Ho 2 .- Identify the alternative hypothesis H1 3 .- Calculate the hypothesis test Z.
Example • Null hypothesis: • when comparing two independent proportions of populations is usually the two proportions are equal. • Ho: π1 = π2 • It is as if the difference in the proportions of the two populations is 0. • Ho: π1 - π2 = 0 • Alternative hypothesis: • is usually that the two proportions are not equal. • H1: π1 ≠ π2 • This is the same as the difference in proportions is not equal to zero. • H1: π1 - π2 ≠ 0
Z statistic test • The general formula for the Z test is the same as for the difference in two means. (p1-p2) – 0 z= -------------- SE(p1-p2) • When the null hypothesis is that the difference in two proportions is zero estimate: (p1-p2) – 0 p1-p2 z= -------------- = -------- SE (p1-p2) SE (p1-p2)
Example • 0.92 success for fosfomycin / trometamol and 0.61 for trimethoprim / sulfamethoxazole SE = 0.019 (p1-p2) – 0 0.31 - 0 z= -------------- = -----------= 16.32 SE(p1-p2) 0.019 P<0.05
Bibliografía • 1.- Last JM. A dictionary of epidemiology. New York, 4ª ed. Oxford University Press, 2001:173. • 2.- Kirkwood BR. Essentials of medical ststistics. Oxford, Blackwell Science, 1988: 1-4. • 3.- Altman DG. Practical statistics for medical research. Boca Ratón, Chapman & Hall/ CRC; 1991: 1-9.