Statistical tests for Quantitative variables (z-test & t-test) BY Dr.Shaikh Shaffi Ahamed Ph.D.,

1. Statistical tests for Quantitative variables (z-test & t-test) BY Dr.Shaikh Shaffi Ahamed Ph.D., Associate Professor Dept. of Family & Community Medicine College of Medicine, King Saud University

2. Choosing the appropriate Statistical test • Based on the three aspects of the data • Types of variables • Number of groups being compared & • Sample size

3. Statistical Tests Z-test: Study variable: Qualitative Outcome variable: Quantitative or Qualitative Comparison: Sample mean with population mean & sample proportion with population proportion; two sample means or two sample proportions Sample size: larger in each group(>30) & standard deviation is known Student’s t-test: Study variable: Qualitative Outcome variable: Quantitative Comparison: sample mean with population mean; two means (independent samples); paired samples. Sample size: each group <30 ( can be used even for large sample size)

4. Test for sample proportion with population proportion Problem In an otological examination of school children, out of 146 children examined 21 were found to have some type of otological abnormalities. Does it confirm with the statement that 20% of the school children have otological abnormalities? a . Question to be answered: Is the sample taken from a population of children with 20% otological abnormality b. Null hypothesis : The sample has come from a population with 20% otological abnormal children

5. Test for sample prop. with population prop. P – Population. Prop. p- sample prop. n- number of samples c. Test statistics d.Comparison with theoritical value Z ~ N (0,1); Z 0.05 = 1.96 The prob. of observing a value equal to or greater than 1.69 by chance is more than 5%. We therefore do not reject the Null Hypothesis e. Inference There is a evidence to show that the sample is taken from a population of children with 20% abnormalities

6. Comparison of two sample proportions Problem In a community survey, among 246 town school children, 36 were found with conductive hearing loss and among 349 village school children 61 were found with conductive hearing loss. Does this data, present any evidence that conductive hearing loss is as common among town children as among village children?

7. Comparison of two sample proportions a. Question to be answered: Is there any difference in the proportion of hearing loss between children living in town and village? Given data sample 1 sample 2 size 246 342 hearing loss 36 61 % hearing loss 14.6 % 17.5% b. Null Hypothesis There is no difference between the proportions of conductive hearing loss cases among the town children and among the village children

8. Comparison of two sample proportions c. Test statistics q= 1- p p1, p2 are sample proportions, n1,n2 are subjects in sample 1 & 2

9. Comparison of two sample proportions d. Comparison with theoretical value Z ~ N (0,1); Z 0.05 = 1.96 The prob. of observing a value equal to or greater than 1.81 by chance is more than 5%. We therefore do not reject the Null Hypothesis e. Inference There is no evidence to show that the two sample proportions are statistically significantly different. That is, there is no statistically significant difference in the proportion of hearing loss between village and town, school children.

10. Exercise Only: Diet Only: measure of variability = [(0.633)2 + (0.540)2] = 0.83 sample mean= 4.1 kg sample standard deviation = 3.7 kg sample size = n = 47 standard error = SEM2 = 3.7/ Ö47 = 0.540 sample mean= 5.9 kg sample standard deviation = 4.1 kg sample size = n = 42 standard error = SEM1 = 4.1/ Ö42 = 0.633 Comparison of two sample meansExample : Weight Loss for Diet vs Exercise Did dieters lose more fat than the exercisers?

11. Example : Weight Loss for Diet vs Exercise Null hypothesis:No difference in average fat lost in population for two methods. Population mean difference is zero. Step 1. Determine the null and alternative hypotheses. Alternative hypothesis:There is a difference in average fat lost in population for two methods. Population mean difference is not zero. Step 2. Sampling distribution: Normal distribution (z-test) Step 3. Assumptions of test statistic ( sample size > 30 in each group) Step 4. Collect and summarize data into a test statistic. The sample mean difference = 5.9 – 4.1 = 1.8 kg and the standard error of the difference is 0.83. So the test statistic: z = 1.8 – 0= 2.17 0.83

12. Example : Weight Loss for Diet vs Exercise Recall the alternative hypothesis was two-sided. p-value = 2  [proportion of bell-shaped curve above 2.17] Z-test table => proportion is about 2  0.015 = 0.03. Step 5. Determine the p-value. Step 6. Make a decision. The p-value of 0.03 is less than or equal to 0.05, so … • If really no difference between dieting and exercise as fat loss methods, would see such an extreme result only 3% of the time, or 3 times out of 100. • Prefer to believe truth does not lie with null hypothesis. We conclude that there is a statistically significant difference between average fat loss for the two methods.

13. Student’s t-test • Test for single mean • Whether the sample mean is equal to the predefined population mean ? • 2. Test for difference in means • Whether the CD4 level of patients taking treatment A is equal to CD4 level of patients taking treatment B ? • 3. Test for paired observation • Whether the treatment conferred any significant benefit ?

14. Steps for test for single mean • Questioned to be answered • Is the Mean weight of the sample of 20 rats is 24 mg? • N=20, =21.0 mg, sd=5.91 , =24.0 mg • 2. Null Hypothesis • The mean weight of rats is 24 mg. That is, The sample mean is equal to population mean. • 3. Test statistics --- t (n-1) df • 4. Comparison with theoretical value • if tab t (n-1) < cal t (n-1) reject Ho, • if tab t (n-1) > cal t (n-1) accept Ho, • 5. Inference

15. t –test for single mean • Test statistics n=20, =21.0 mg, sd=5.91 , =24.0 mg t = t .05, 19 = 2.093 Accept H0 if t < 2.093 Reject H0 if t >= 2.093 Inference : We reject Ho, and conclude that the data is not providing enough evidence, that the sample is taken from the population with mean weight of 24 gm

17. Lean 6.1 7.0 7.5 7.5 5.5 7.6 7.9 8.1 8.1 8.1 8.4 10.2 10.9 t-test for difference in means Given below are the 24 hrs total energy expenditure (MJ/day) in groups of lean and obese women. Examine whether the obese women’s mean energy expenditure is significantly higher ?. Obese 8.8 9.2 9.2 9.7 9.7 10.0 11.5 11.8 12.8

18. Null Hypothesis Obese women’s mean energy expenditure is equal to the lean women’s energy expenditure. Data Summary lean Obese N 13 9 8.10 10.30 S 1.38 1.25

19. Solution • H0: m1 - m2 = 0(m1 = m2) • H1: m1 - m2 ¹ 0 (m1¹ m2) • a = 0.05 • df = 13 + 9 - 2 = 20 • Critical Value(s): Reject H Reject H 0 0 .025 .025 t -2.086 0 2.086

20. Calculating the Test Statistic: Hypothesized Difference (usually zero when testing for equal means) • Compute the Test Statistic: _ _ ) ( ) ( - m - m - X X 1 2 1 2 t = n1 n2 df = n + n - 2 1 2 ( ) ( ) × 2 × 2 + n - 1 S n - 1 S 2 1 1 2 2 S = ( ) ( ) P n - 1 + n - 1 1 2

21. Developing the Pooled-Variance t Test • Calculate the Pooled Sample Variances as an Estimate of the Common Populations Variance: = Pooled-Variance = Size of Sample 1 = Variance of Sample 1 = Size of Sample 2 = Variance of sample 2

22. First, estimate the common variance as a weighted average of the two sample variances using the degrees of freedom as weights ( ) ( ) 2 2 n - 1 × S + n - 1 × S 1 1 2 2 2 S = ( ) ( ) P n - 1 + n - 1 1 2 ( ) ( ) 2 2 × . 13 - 1 × 38 + 9 - 1 1 . 25 1 . = 765 = 1 ) ( ) ( 13 - + 9 - 1 1

23. Calculating the Test Statistic: ( ) ) ( ) ( X X - - m - m - 10.3 - 0 8.1 1 2 1 2 = 3.82 t = = 2 1 + 13 1 9 S 1 . 76 P n n 1 2 tab t 9+13-2 =20 dff = t 0.05,20 =2.086

25. T-test for difference in means Inference : The cal t (3.82) is higher than tab t at 0.05, 20. ie 2.086 . This implies that there is a evidence that the mean energy expenditure in obese group is significantly (p<0.05) higher than that of lean group

26. Suppose we want to test the effectiveness of a program designed to increase scores on the quantitative section of the Graduate Record Exam (GRE). We test the program on a group of 8 students. Prior to entering the program, each student takes a practice quantitative GRE; after completing the program, each student takes another practice exam. Based on their performance, was the program effective? Example

27. Each subject contributes 2 scores: repeated measures design

28. Can represent each student with a single score: the difference (D) between the scores

29. Approach: test the effectiveness of program by testing significance of D Null hypothesis: There is no difference in the scores of before and after program Alternative hypothesis: program is effective → scores after program will be higher than scores before program → average D will be greater than zero H0: µD = 0 H1: µD > 0

30. So, need to know ∑D and ∑D2:

31. Recall that for single samples: For related samples: where: and

32. Mean of D: Standard deviation of D: Standard error:

33. Under H0, µD = 0, so: From Table B.2: for α = 0.05, one-tailed, with df = 7, t critical = 1.895 2.714 > 1.895 → reject H0 The program is effective.

35. Z- value & t-Value “Z and t” are the measures of: How difficult is it to believe the null hypothesis? High z & t values Difficult to believe the null hypothesis - accept that there is a real difference. Low z & t values Easy to believe the null hypothesis - have not proved any difference.

36. In conclusion ! Z-test will be used for both categorical(qualitative) and quantitative outcome variables. Student’s t-test will be used for only quantitative outcome variables.