Chapter 14 Tests of Hypotheses Based on Count Data 14.2 Tests concerning proportions (large samples) 14.3 Differences between proportions 14.4 The analysis of an r x c table 14.2 Tests concerning proportions (large samples) np>5; n(1-p)>5 n independent trials; X=# of successes p=probability of a success Estimate: p X n Tests of Hypotheses Null H0: p=p0 Possible Alternatives: HA: pp0 HA: pp0
Test Statistics Under H0, p=p0, and p(1 p) p0 (1 p0 ) p n n Statistic: p p0 p p0 z p p0 (1 p0 ) n is approximately standard normal under H0 . Reject H0 if z is too far from 0 in either direction.
Rejection Regions Alternative Hypotheses HA: p>p0 HA: pz z<-z z>z/2 or z<-z/2 Equivalent Form:
p p0 X np0 n z * p0 (1 p0 ) n np0 (1 p0 ) n Example 14.1 H0: p=0.75 vs HA: p0.75 =0.05 n=300 x=206 Reject H0 if z<-1.96 or z>1.96 p X 206
0.68667 n 300 Observed z value 0.68667 0.75 z 2.5 0.75(1 0.75) 300 or 206 225 z 2.5 300(0.75)(1 0.75) Conclusion: reject H0 since z<-1.96 P(z<-2.5 or z>2.5)=0.0124
Is the coin balanced one? =0.05 Solution: H0: p=0.50 vs HA: p0.50 45 p 0.45 100 Enough Evidence to Reject H0? Critical value z0.025=1.96 Reject H0 if z>1.96 or z<-1.96 45 100(0.50) 5 z 1 5 100(0.50)(1 0.50) Conclusion: accept H0 Another example
The following table is for a certain screening test Truth = surgical biopsy FNA status Result Positive Results Negative Total sensitivity Cancer Present 140 Cancer Absent 80 10
910 150 990 True positive 140 0.93 True Positives False Negatives 150 Total 220 920 1140 Test to see if the sensitivity of the screening test is less than 97%. Hypothesis H : p p .97 0
0 Ha : p p0 .97 Test statistic estimated proportion-prestated proportion z standard error of the estimated proportion p p0 p p0 140 150 .97 2.6325 SE p p0 (1 p0 ) .97 (1 .97) 150 n What is the conclusion? Check p-value when z=-2.6325, p-value =
0.004 Conclusion: we can reject the null hypothesis at level 0.05. One word of caution about sample size: If we decrease the sample size by a factor of 10, Truth = surgical biopsy FNA status Cancer Present Cancer Absent Total Result Positive
114 True positive 14 0.93 True Positives False Negatives 15 And if we try to use the z-test, z estimated proportion-prestated proportion standard error of the estimated proportion p p0 p p0 14 15 .97 0.8324 SE p
p0 (1 p0 ) .97 (1 .97) 15 n P-value is greater than 0.05 for sure (p=0.2026). So we cannot reach the same conclusion. And this is wrong! So for test concerning proportions We want np>5; n(1-p)>5 14.3 Differences Between Proportions Two drugs (two treatments) p1 =percentage of patients recovered after taking drug 1 p2 =percentage of patients recovered after
taking drug 2 Compare effectiveness of two drugs Tests of Hypotheses Null H0: p1=p2 (p1-p2 =0) Possible Alternatives: HA: p1p2 HA: p1p2 Compare Two Proportions Drug 1: n1 patients, x1 recovered Drug 2: n2 patients, x2 recovered Estimates: x x 1 1 ; p 2 2 p Statistic for test: n1 n2
z 1 p 2 p 1and 2 p p If we did this study over and over drew a histogram of the resulting values of , that histogram or distribution would have standard deviation p1 p 2 p1 p 2 Estimating the Standard Error
Under H0, p1=p2=p. So p2 p p2 p2 p (1n p ) 1 1 2 1 2 1 1 1 1 p (1 p )( ) n1 n2 Estimate the common p by
n1 p 1 n2 p 2 x1 x2 p n1 n2 n1 n2 p 2 (1 p 2 ) n2 So put them together p1 p 2 z 1 1 p (1 p )( ) n1 n2 Example 12.3 Two sided test: H0: p1=p2 vs HA: p1p2 n1=80, x1=56 n2=80, x2=38 p 1 0.7
p 2 0.475 p 56 38 0.5875 80 80 Two Tailed Test Observed z-value: z 0.7 0.475 0.225 2.88 0.078 1 1 0.5875 0.4125( ) 80 80
Critical value for two-tailed test: 1.96 Conclusion: Reject H0 since |z|>1.96 Rejection Regions Alternative Hypotheses HA: p1>p2 HA: p1z z<-z z>z/2 or
z<-z/2 P-value of the previous example P-value=P(z<-2.88)+P(z>2.88)=2*0.004 So not only we can reject H0 at 0.05 level, we can also reject at 0.01 level. 14.4 The analysis of an r x c table Recall Example 12.3 Two sided test; n1=80, x1=56 H0: p1=p2 vs HA: p1p2 n2=80, x2=38 We can put this into a 2x2 table and the question now becomes is there a relationship between treatment and outcome? We will come back to this example after we introduce 2x2 tables and chi-square test. recover Not recover
trt1 56 24 80 trt2 38 42 80 94 66 2x2 Contingency Table The table shows the data from a study of 91 patients who had a myocardial infarction (Snow 1965). One variable is treatment (propranolol versus a placebo), and the other is outcome (survival for at least 28 days versus death within 28 days). OUTCOME Survival for at
Death least 28 days Treatment Propranolol 38 Total 7 45 Placebo 29 17 46 Total
67 24 91 Hypotheses for Two-way Tables The hypotheses for two-way tables are very broad stroke. The null hypothesis H0 is simply that there is no association between the row and column variable. The alternative hypothesis Ha is that there is an association between the two variables. It doesnt specify a particular direction and cant really be described as one-sided or twosided. Hypothesis statement in Our Example Null hypothesis: the method of treating the myocardial infarction patients did not influence the proportion of patients who survived for at least 28 days. The alternative hypothesis is that the outcome
(survival or death) depended on the treatment, meaning that the outcomes was the dependent variable and the treatment was the independent variable. Calculation of Expected Cell Count To test the null hypothesis, we compare the observed cell counts (or frequencies) to the expected cell counts (also called the expected frequencies) Row1 Total Column1 Total E1,1 Study Total The process of comparing the observed counts with the expected counts is called a goodness-of-fit test. (If the chisquare value is small, the fit is good and the null hypothesis is not rejected.) Observed cell counts
Treatment OUTCOME Survival for at Death least 28 days Propranolol 38 Placebo Total 29 7 17 67 24 Total
45 46 91 Expected cell counts OUTCOME Survival for at Death least 28 days Treatment Propranolol 33.13 Placebo Total 33.87 11.87 12.13
67 24 Total 45 46 91 The Chi-Square ( 2) Test Statistic The chi-square statistic is a measure of how much the observed cell counts in a two-way table differ from the expected cell counts. It can be used for tables larger than 2 x 2, if the average of the expected cell counts is > 5 and the smallest expected cell count is > 1; and for 2 x 2 tables when all 4 expected cell counts are > 5. The formula is: 2 = (observed count expected count)2/expected countexpected count Degrees of freedom (df) = (r 1) x (c 1) Where observed is an observed sample count and expected is the computed expected cell count for the same cell, r is the number
of rows, c is the number of columns, and the sum () is over all the r x c cells in the table (these do not include the total cells). The Chi-Square ( 2) Test Statistic Calculation of the chi - square ( 2 ) value (O E ) 2 E (38 33.13) 2 (7 11 .87) 2 ( 29 33.87) 2 (17 12.13) 2 33.13 11 .87 33.87 12.13 ( 4.87) 2 ( 4.87) 2 ( 4.87) 2 ( 4.87) 2
Interpretation : The results noted in this 2 2 table are statistically significant. That is, it is highly probable (only 1 chance in about 50 of being wrong) that the investigator can reject the null hypothesis of independence and accept the alternative hypothesis that propranolol does affect the outcome of myocardialinfarction. Example: Patient Compliance w/expected count Rx In a study of 100 patients with hypertension, 50 were randomly allocated to a group prescribed 10 mg lisinopril to be taken once daily, while the other 50 patients were prescribed 5 mg lisinopril to be taken twice daily. At the end of the 60 day study period the patients returned their remaining medication to the research pharmacy. The pharmacy then counted the remaining pills and classified each patient as < 95% or 95%+ compliant with their prescription. The two-way table for Compliance and Treatment was: Compliance Treatment 10 mg Daily 5 mg bid
Total 95%+ 460 400 860 < 95% 40 100 140 Total 500 500 The expected cell counts were: 1000 Treatment Compliance 10 mg Daily 5 mg bid Total 95%+ 430 430
860 < 95% 70 70 140 Total 500 500 2 = 29.9, df = (2-1)*(2-1) = 1, P-value <0.001 1000 If we use the two sample test for proportion 460 400 860 p1 , p 2 , p 500 500 1000 p1 p 2
z 5.46812 1 1 p (1 p )( ) n1 n2 5.468122 29.9 z 2 2 (1) The 2 and z Test Statistics The comparison of the proportions of successes in two populations leads to a 2 x 2 table, so the population proportions can be compared either using the 2 test or the two-sample z test . It really doesnt matter, because they always give exactly the same result, because the 2 is equal to the square of the z statistic and the chi-square with one degree of freedom 2(1) critical values are equal to the squares of the corresponding z critical values. A P-value for the 2 x 2 2 can be found by calculating the square root of the chi-square, looking that up in Table for P(Z > z) and multiplying by 2, because the chi-square always tests the twosided alternative. For a 2 x 2 table with a one-sided alternative hypothesis the twosample z statistic would need to be used. To test more than two populations the chi-square must be used The chi-square is the one most often seen in the literature
Summary: Computations for Two-way Tables 1. create the table, including observed cell counts, column and row totals. 2. Find the expected cell counts. Determine if a 2 test is appropriate Calculate the 2 statistic and number of degrees of freedom 3. Find the approximate P-value use Table III chi-square table to find the approximate P-value or use z-table and find the two-tailed p-value if it is 2 x 2. 4. Draw conclusions about the association between the row and column variables. Yates Correction for Continuity The chi-square test is based on the normal approximation of the binomial distribution (discrete), many statisticians believe a correction for continuity is needed. 2 (| O
E | 0 . 5 ) 2 Yates E It makes little difference if the numbers in the table are large, but in tables with small numbers it is worth doing. It reduces the size of the chi-square value and so reduces the chance of finding a statistically significant difference, so that correction for continuity makes the test more conservative. What do we do if the expected
values in any of the cells in a 2x2 table is below 5? For example, a sample of teenagers might be divided into male and female on the one hand, and those that are and are not currently dieting on the other. We hypothesize, perhaps, that the proportion of dieting individuals is higher among the women than among the men, and we want to test whether any difference of proportions that we observe is significant. The data might look like this: men women tota l dieting 1 9 10 not dieting
11 3 14 The question we ask about these data is: knowing that 10 of these 24 teenagers are dieters, what is the probability that these 10 dieters would be so unevenly distributed between the girls and the boys? If we were to choose 10 of the teenagers at random, what is the probability that 9 of them would be among the 12 girls, and only 1 from among the 12 boys? --Hypergeometric distribution! --Fishers exact test uses hypergeometric distribution to calculate the exact probability of obtaining such set of the values. Fishers exact test Before we proceed with the Fisher test, we first introduce some notation. We represent the cells by the letters a, b, c and d, call the totals across rows
and columns marginal totals, and represent the grand total by n. So the table now looks like this: men women total dieting a b a+b not dieting c d c+d totals
a+c b+d n men women total dieting a b a+b not dieting c
d c+d totals a+c b+d n Fisher showed that the probability of obtaining any such set of values was given by the hypergeometric distribution: In our example As extreme as observed men
women total dieting 1 9 10 not dieting 11 3 totals 12
12 More extreme than observed men women total dieting 0 10 10 14
not dieting 12 2 14 24 totals 12 12 24 10!14!12!12! p 0.00134
24!1!9!11!3! 10!14!12!12! p 0.00003 24!0!10!12!2! Recall that p-value is the probability of observing data as extreme or more extreme if the null hypothesis is true. So the p-value is this problem is 0.00137. The fisher Exact Probability Test Used when one or more of the expected counts in a contingency table is small (<2). Fisher's Exact Test is based on exact probabilities from a specific distribution (the hypergeometric distribution). There's really no lower bound on the amount of data that is needed for Fisher's Exact Test. You can use Fisher's Exact Test when one of the cells in your table has a zero in it. Fisher's Exact Test is also very useful for highly imbalanced tables. If one or two of the cells in a two by two table have numbers in the thousands and one or two of the other cells has numbers less
than 5, you can still use Fisher's Exact Test. Fisher's Exact Test has no formal test statistic and no critical value, and it only gives you a p-value.
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