n318b winter 2002 nursing statistics hypothesis and inference tests, type i and ii errors, p-values,...

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

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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


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“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 !


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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)


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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”


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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


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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.


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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


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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


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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


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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)


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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


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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)


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10 minute break !


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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!


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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)


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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?


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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


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Part 2: Application to the

Assigned Readings


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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


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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


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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)


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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?


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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)


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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


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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?


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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%)?


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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)


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Q4. What do the CI’s in this table tell you?

Row 1, Symptom 95% CI = (0.79, 0.86)


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)


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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


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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)

n318b winter 2002 nursing statistics hypothesis and inference tests, type i and ii errors, p-values,...

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