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N318b Winter 2002 Nursing Statistics
Hypothesis and Inference tests, Type I and II errors, p-values,
Confidence Intervals
Lecture 5
Nur 318b 2002 Lecture 4: page 2
School ofNursing
Institute for Work & Health
Today’s Class Hypothesis testing & inference Types of errors (of inference) P-values << 10 min break >> Confidence intervals Applying knowledge to assigned readings
Gulick (1995); Birenbaum et al. (1996)
Followed by small groups from 12-2 PM
Nur 318b 2002 Lecture 4: page 3
School ofNursing
Institute for Work & Health
“In Group” Session
Focuses on two assigned readings.Q1 is a review of descriptive dataQ2 discusses hypothesis testingQ3 covers confidence intervals
Key points from the readings will be covered in the 2nd part of the lecture !
Nur 318b 2002 Lecture 4: page 4
School ofNursing
Institute for Work & Health
A Quick Review from Last Week
Normal distributionMean = 0, SD = 1A mathematical solution for “reality”Z-scores convert SD to probability
Central Limit TheoremMeans are also normally distributedSE (of means) similar to SD (of data)(For samples of about 25 or more)
Nur 318b 2002 Lecture 4: page 5
School ofNursing
Institute for Work & Health
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”
Nur 318b 2002 Lecture 4: page 6
School ofNursing
Institute for Work & Health
Hypothesis testing - cont’d
How do you do it?
H0 : sample mean = population mean
Start with a “null” hypothesis since no single study can never “prove” anything
Ha : sample mean population mean
but we can “reject” notion of no effect
Nur 318b 2002 Lecture 4: page 7
School ofNursing
Institute for Work & Health
Hypothesis testing - cont’d
5 steps to setting up test:
1. State null (H0 ) & alternative (Ha ) hypotheses
4. Calculate test statistic (e.g. Z-score)
2. Choose statistic you will test
3. Set “chance” error level (alpha level)
5. Can H0 be rejected (i.e. p < )? Make your conclusions about the data.
Nur 318b 2002 Lecture 4: page 8
School ofNursing
Institute for Work & Health
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. H0: = 110 versus Ha 110
3. Set “chance” error level ( = 0.05)
2. We will test the (sample) mean BP
Nur 318b 2002 Lecture 4: page 9
School ofNursing
Institute for Work & Health
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
Hypothesis testing - example4. Z-score calculated as before
5. Compare Z-score to critical value
Nur 318b 2002 Lecture 4: page 10
School ofNursing
Institute for Work & Health
1-tailed versus 2-tailed testsWas 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
Nur 318b 2002 Lecture 4: page 11
School ofNursing
Institute for Work & Health
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)
Nur 318b 2002 Lecture 4: page 12
School ofNursing
Institute for Work & Health
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
Nur 318b 2002 Lecture 4: page 13
School ofNursing
Institute for Work & Health
Significance LevelBoth 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)
Nur 318b 2002 Lecture 4: page 14
School ofNursing
Institute for Work & Health
10 minute break !
Nur 318b 2002 Lecture 4: page 15
School ofNursing
Institute for Work & Health
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!
Nur 318b 2002 Lecture 4: page 16
School ofNursing
Institute for Work & Health
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)
Nur 318b 2002 Lecture 4: page 17
School ofNursing
Institute for Work & Health
Confidence limits – example
For our BP point estimate of 113 mmHG, with n=100, and SD=15 mmHg
For a 95% CI: 1.96 SE = 113 1.96 (15/ 100)= 113 2.94= (110.06, 115.94)
What does this mean?
Nur 318b 2002 Lecture 4: page 18
School ofNursing
Institute for Work & Health
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
Nur 318b 2002 Lecture 4: page 19
School ofNursing
Institute for Work & Health
Part 2: Application to the
Assigned Readings
Nur 318b 2002 Lecture 4: page 20
School ofNursing
Institute for Work & Health
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
Nur 318b 2002 Lecture 4: page 21
School ofNursing
Institute for Work & Health
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
Nur 318b 2002 Lecture 4: page 22
School ofNursing
Institute for Work & Health
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)
Nur 318b 2002 Lecture 4: page 23
School ofNursing
Institute for Work & Health
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?
Nur 318b 2002 Lecture 4: page 24
School ofNursing
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Table 1 … cont’d
Hard to tell without median, but range may give some insighte.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)
Nur 318b 2002 Lecture 4: page 25
School ofNursing
Institute for Work & Health
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
Nur 318b 2002 Lecture 4: page 26
School ofNursing
Institute for Work & Health
Q1. How did the authors make use of CI’s? (Hint – see 1st 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?
Nur 318b 2002 Lecture 4: page 27
School ofNursing
Institute for Work & Health
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%)?
Nur 318b 2002 Lecture 4: page 28
School ofNursing
Institute for Work & Health
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)
Nur 318b 2002 Lecture 4: page 29
School ofNursing
Institute for Work & Health
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)
Nur 318b 2002 Lecture 4: page 30
School ofNursing
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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
Nur 318b 2002 Lecture 4: page 31
School ofNursing
Institute for Work & Health
Next Week - Lecture 6: Parametric and non-parametric
tests; Chi square (2) test
For next week’s class please review:1. Page 15 in syllabus2. Textbook Chapter 4, pages 97-1073. Syllabus paper:
Turk et al. (1995)