1. N318b Winter 2002 Nursing Statistics Hypothesis and Inference tests, Type I and II errors, p-values, Confidence Intervals Lecture 5

2. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 2 Today’s Class  Hypothesis testing & inference  Types of errors (of inference)  P-values  >  Confidence intervals  Applying knowledge to assigned readings Gulick (1995); Birenbaum et al. (1996) Followed by small groups from 12-2 PM

3. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 3 “In Group” Session Focuses on two assigned readings. Q1 is a review of descriptive data Q2 discusses hypothesis testing Q3 covers confidence intervals Key points from the readings will be covered in the 2 nd part of the lecture !

4. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 4 A Quick Review from Last Week Normal distribution Mean = 0, SD = 1 A mathematical solution for “reality” Z-scores convert SD to probability Central Limit Theorem Means are also normally distributed SE (of means) similar to SD (of data) (For samples of about 25 or more)

5. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 5 Hypothesis testing Why do it? What is it? Need to be able to draw conclusions (i.e. inferences) about samples that we observe. Is what we see “real”? A process whereby “observed” data (e.g. mean) compared to “expected”

6. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 6 Hypothesis testing - cont’d How do you do it? H 0 : sample mean = population mean Start with a “null” hypothesis since no single study can never “prove” anything H a : sample mean  population mean but we can “reject” notion of no effect

7. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 7 Hypothesis testing - cont’d 5 steps to setting up test: 1. State null (H 0 ) & alternative (H a ) hypotheses 4. Calculate test statistic (e.g. Z-score) 2. Choose statistic you will test 3. Set “chance” error level (alpha level) 5. Can H 0 be rejected (i.e. p <  )? Make your conclusions about the data.

8. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 8 Hypothesis testing - example Test hypothesis that sample mean systolic BP of 113 mmHG (n=100) differs from the population mean of 110 mmHg (assume SD=15 mmHG as from last lecture) 1. H 0:  = 110 versus H a   110 3. Set “chance” error level (  = 0.05) 2. We will test the (sample) mean BP

9. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 9 Z = ------ 113 - 110 = --------- 15 / 10 = 2.0 Critical region for  =0.05 is +/- 1.96 (From Z-score Table in Appendix A)  we can reject null hypothesis (we can publish!) Sample mean Z-score exceeds critical value  -   / n / n Hypothesis testing - example 4. Z-score calculated as before 5. Compare Z-score to critical value

10. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 10 1-tailed versus 2-tailed tests Was our example a 1- or 2-tailed test? Need to look at our hypothesis, which states only that the sample mean is different – does not specify a direction! Thus we used a 2-tailed test What do the “tails” refer to? See Figure 3.3 page 83 and Figure 3.4 page 83-84 of textbook

11. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 11 Our studies are never perfect and never generate error-free results, thus mistakes can be made regarding study conclusions errors classified in two ways: Types of (inference) errors Type II – accept null hypothesis when a real effect (e.g. difference) is present Type I – rejection of null hypothesis when there is no real effect (e.g. no difference)

12. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 12 Types of (inference) errors Type II – also called beta (  ) error since it is associated with the power of the study (often result of sample size being too small) Type I – also called alpha (  ) error since it is associated with the critical value

13. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 13 Significance Level Both types or errors relate to significance levels or p-values: An expression of the probability of observing your study results by (random) chance alone (i.e. if you sampled from overall population at random what is the likelihood you would get same result?) By convention only (i.e. arbitrary) the accepted level is p < 0.05 Smaller is better (i.e. higher significance)

14. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 14 10 minute break !

15. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 15 Confidence limits Always have error associated with sample statistics (point estimates) – e.g. mean Would be nice to have a way of expressing statistically the “precision” of estimates Can use the theory underlying central limit theorem, Z-scores and normal distribution to do this by putting upper and lower bounds on point estimate!

16. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 16 Confidence limits – cont’d Recall that 95% of estimates for single values from a normal distribution will lie between 1.96 SD on either side of mean For mean values, we substitute SE (standard error) for SD thus 95% of sample means will lie between 1.96 SE on either side of mean For a 95% CI:   1.96 SE =  1.96 (SD/  n)

17. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 17 Confidence limits – example For our BP point estimate of 113 mmHG, with n=100, and SD=15 mmHg For a 95% CI: 0  1.96 SE = 113  1.96 (15/  100) = 113  2.94 = (110.06, 115.94) What does this mean?

18. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 18 Confidence limits – example What does this mean? We can expect the sample mean to fall within this range in 95% of the samples that are taken Does NOT mean there is a 95% chance that the true mean is between 110.06 and 115.94

19. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 19 Part 2: Application to the Assigned Readings

20. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 20 Gulick et al. (1995) Quick summary of the paper: – a cross-sectional study examining coping strategies used by spouses/others (SOS) of people living with MS – 156 MS subjects and 156 SOS subjects were enrolled in the study – related dependency of MS subjects to coping strategies developed by SOS

21. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 21 A question … dependency and coping treated as interval scales (or possibly ordinal?) How were these variables expressed? What were the key study variables? dependency and coping (sub-scales) Dependency = 0-5 score Coping = 0-3 score

22. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 22 A question … cont’d Measures of central tendency and dispersion – e.g. mean, median, mode, SD, range, etc. What statistics best describe interval data ? How can such small scores be interval data? These are multi-item scales expressed on the same scale for comparability (i.e. “raw” scores are often much larger)

23. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 23 A question about Table 1 … Fine gross motor - 118/156 afflicted What was most common problem? How would you describe the data? Presents descriptive statistics for the MS dependency scales (0-5 scoring) Rec/Soc – mean was highest (2.81) What was most serious problem?

24. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 24 Table 1 … cont’d Hard to tell without median, but range may give some insight e.g. for fine gross motor scale has mean value at low end of range (possibly left-skewed?) Were any variables skewed? Which scale was most variable? SD highest relative to mean for fine gross motor (i.e. CV=1.33/1.47=0.90)

25. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 25 Birenbaum et al (1996) Quick summary of the paper: – a prospective study looking at the health effects of a child’s dying on the parents – 48 families entered study during terminal phase of child’s cancer (80 parents) – parents interviewed at four time points (before; 2wks-, 4wks-, 52wks after death) – did not observe a significant reduction in parental health after the loss of a child

26. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 26 Q1. How did the authors make use of CI’s? (Hint – see 1 st paragraph of the Results section) “To compare the means of the current study with normative data, 95% CI’s were used” A question about CI’s … What does “normative” data mean? Why is it useful in this case?

27. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 27 Q2. How did the authors define CI’s? “Confidence limits specify the level of certainty (in this case 95%) with which the [real or true] mean lies between two boundary points” Another question about CI’s … What happens to the size of the interval if the precision level is increased (e.g. 99%) or decreased (e.g. 90%)?

28. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 28 Q3. How do you interpret this table – i.e. what information does it offer? Always look at column and row headings first, then footnotes – be sure to know what each is telling you (refer to text as needed) A question about Table 1 … This table displays the typical format for prospective results – i.e. time on one axis (rows), outcome down the other (columns)

29. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 29 Q4. What do the CI’s in this table tell you? Row 1, Symptom 95% CI = (0.79, 0.86) Table 1 … cont’d What is the point estimate for the mean? What is the “population” or true value?  = 0.82 (SD=0.13)  = 0.84 (SD=0.11)

30. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 30 Table 1 … cont’d Are the “population” and study sample values different from one another? In other words, does 0.84 lie within the 95% CI for the sample mean? YES, thus the sample could have been drawn from same population as reference group, therefore no difference in health between bereaved parents and others

31. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 4: page 31 Next Week - Lecture 6: Parametric and non-parametric tests; Chi square (  2 ) test For next week’s class please review: 1.Page 15 in syllabus 2.Textbook Chapter 4, pages 97-107 3.Syllabus paper: Turk et al. (1995)