1. Biostatistic course Part 10 Inferences from a proportion Dr. Sc. Nicolas Padilla Raygoza Department dof Nursing and Obstetrics Division Health Sciences and Engineering Campus Celaya-Salvatierra University of Guanajuato Mexico

2. Biosketch Medical Doctor by University Autonomous of Guadalajara. Pediatrician by the Mexican Council of Certification on Pediatrics. Postgraduate Diploma on Epidemiology, London School of Hygine 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

3. Competencies The reader will apply a Z test to obtain inferences from a proportion. He (she) will obtain a confidence interval from a proportion.

4. Introduction It is common in health studies, to measure categorical variables: Gender: male or female, Civil status: single, married, widow, divorced, separate, free union. Result of detection of antigen of Entamoeba histolytic: positive or negative.

5. Introduction When we use variables with only two categories, summarize them with proportions or percentages. MedicationCure n % No cure n % Quinfamide (n=89) 79 88.7610 11.24 Nitazoxanide (n=105) 79 75.24 26 24.76

6. Introduction In the study of anti-amoebic treatment, the proportion of children with amebiasis was of 0.277. The proportion of children with amebiasis was 194/700 = 0.277 0.277(1-0.277) Standard error is: √ -------------------- = 0.017 700

7. Notation The notation for population and sample parameters, for proportions are shown. Remember, we use Greek letters for population parameters and Latin letters for the sample. PopulationSample Proportionπp Standard deviation √π(1-π)√P(1-P)

8. Example We we can search the distribution from a binary variable, as gender in children in schools from Celaya, we take a sample of size n from teh population from all children in all schools from Celaya; a proportion, p, from all children in the sample, that they have the characteristics of interest. To obtain conclusions about a proportion from the population, we apply the same methods to obtain inferences about means from samples.

9. Confidence interval 95% from a proportion If the sample size is big,we can calculate a confidence interval for a ´proportion of a sample using the common formulae: Proportion ± 1.96 x SE (proportion) p±1.96 SE (p)

10. Hypothesis test for a proportion If we can evaluate if the proportion has a value, we use a procedure to test hypothesis. 1.- First, we should note the null hypothesis: H o : π = π o 2. Then, we note the alternative hypothesis: H 1 : π ≠ π o 3. We calculate the statistic test.

11. Hypothesis test for a proportion The formulae is the similar that the formulae used with means, but using proportions instead of means. p – π o Z = --------- SE (p) Where π o is the hypothesis proportion and p is the proportion observed in the sample. Z value represent the number or standard errors between the proportions of hypothesis and the observed.

12. Example In the sample of 700 children, we searched amebiasis diagnosis,27.7% was detected the antigen of E. histolytic in feces. The principal researcher was surprised, because he thought that in that area, amebiasis prevalence was 15%.

13. Example We can probe if the observed value is really different than expected value. Null hypothesis: H o : p = π o = 0.15 Alternative hypothesis: H 1 : p ≠ π o If standard error of the proportion is 0.017, Hypothesis test is Z = p – π o / ES (p) = 0.27 – 0.15 / 0.017= 7.05 P-value for Z of 7.05 is <0.05.

14. Example The interpretation is: If the proportion of patients with amebiasis in the population is 15%, the opportunity to observe a proportion from the sample equal or more extreme than 27% is less than 0.05.

15. Small samples The methods to make inferences from a sample from the population, describe in this lecture, only are valid if the sample size is big. A rule to decide if the sample size is sufficiently big for the distributions of the proportions is Normal is: 1. π n > 5 2. (1 – π) n > 5

16. Small samples If the rule is not accomplished, we should calculate the Fisher exact test.

17. 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.