biol2300 biostatistics chapter 9 - boston...

BIOL2300 BiostatisticsChapter 9

Hypothesis testing in case of 2 proportions, 2 means, and 2

variances (or standard deviations)

Test statistic z when H0 is “p1=p2”

Confidence interval for difference p1-p2 between proportions in two populations

Test statistic z when H0 is “p1 = p2+c”

Justification

P-value versus test statistic in rejection region

Matched vs unmatched pairs

• Two samples are dependent if members of one sample can be used to determine members of the second sample. Such samples are said to be matched.

• Two samples are independent if members selected in first sample are chosen randomly, with no relation and no regard to members in the second sample. Such samples are said to be unmatched.

Answer: B

• The samples are independent. Knowledge of who is in one class gives no information about who is in the other class -- i.e. independent, not matched pairs.

Answer: A

• Matched pairs, since it is the SAME person before and after application of a medication.

Test statistic for difference in means(unmatched case)

Confidence interval for difference of means

Test statistic for difference in means(matched pairs)

Odds ratio (OR) and risk ratio (RR)

HYPOTHESIS TESTING FOR VARIANCES OF TWO SAMPLES

Requirements• The two populations are independent of each other

(i.e. samples are not matched or paired and the two populations are unrelated in any manner)

• The two populations are each normally distributed. WARNING: Methods for testing assertions about variances from two populations are extremely sensitive to departures from normality. If test statistic is in the rejection region, then this could mean that the null hypothesis is not true OR that there was an unobserved departure from normality.

F-distribution

Test the indicated claim about the variances or standard deviations of two populations. Assume that the populations are normally

distributed. Assume that the two samples are independent and that they have been randomly selected.

• A random sample of 16 women resulted in blood pressure levels with a standard deviation of 22.6 mm Hg. A random sample of 17 men resulted in blood pressure levels with a standard deviation of 21 mm Hg. Use a 0.025 significance level to test the claim that blood pressure levels for women have a larger variance than those for men.

Using EXCEL functions• F.INV(0.975,15,16)=2.787517572Critical value F such that area from –infty to F is ®This convention is OPPOSITE that of TABLE A-5•• F.DIST(2.7875,15,16,TRUE)=0.974999322Returns cumulative area from –infty to critical value

• F.DIST.RT(2.7875,15,16)=0.02500067Returns right tailed area from critical value to +infty

EXCEL function for T-test• T.TEST(array1,array2,tails,type) returns the

p-value for 1-tailed or 2-tailed T-test. Here “type” indicate type of T-test to perform: 1 mean a PAIRED test, 2 means a two-sample, equal variance test, 3 performs a two-sample unequal variance test.

• If there are N subjects, with measurements before/after then use 1; otherwise generally use 3

EXCEL functions for equality hypothesis testing of two variances

• F.TEST(array1, array2) returns the two-tailed probability that the variances in array1 and array2 are not significantly different

• The following function computes the critical value. WARNING: Excel formalism different than tables in book! See next page as well.

• F.INV(®,df1,df2) = x if and only if F.DIST(x,df1,df2,TRUE) = ® . For example, F.DIST(5,9,11,TRUE)=0.992534009 F.INV(0.992534009,9,11)=5

EXCEL functions for equality hypothesis testing of two variances

• F.DIST(x,df1,df2,TRUE) returns P( F<x ), where F is a random variable that has an F distribution with degrees of freedom df1 for variance (s1)2 and variance (s2)2, where s1 ¸ s2 . NOTE: not F.DIST here, but other functions are F.

• F.DIST.RT(x,df1,df2) returns P( F>x ), where F is a random variable that has an F distribution with df1 and df2