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- Introduction to Biostatistics II
- Jane L. Meza, Ph.D.
- October 24, 2005

Outline

- Hypothesis testing
- Comparing 2 groups
- Paired t-test
- 2 Independent Samples t-test
- Wilcoxon Signed Ranks test
- Wilcoxon Rank Sum test
- Comparing 3 or more groups
- ANOVA
- One-Way
- Bonferroni Comparisons
- Repeated Measures
- Kruskal-Wallis
- Chi-square
- Regression
- Linear Correlation
- Linear Regression

Deck of Cards

- If you randomly select a card, what is the

probability the card is red? - If we draw 10 cards, how many of the 10 cards do

we expect to be red? - Are we guaranteed that 5 of the cards will be red?

Deck of Cards Experiment

- Suppose we draw 10 cards from a deck of 52 cards,

and all 10 cards are red. - Is it possible that we could draw 10 red cards

from a standard deck of cards? - Is it very likely that we could draw 10 red cards

from a standard deck of cards? - We have conflicting information we assumed that

50 of the cards were red, but in our sample 100

of the cards were red. What should we conclude?

Experiment

- Why did you make that conclusion?
- What assumptions are you making?
- Is there a possibility that your conclusion is

incorrect?

Hypothesis Testing

- Start with an assumption (Null Hypothesis)
- 50 of the cards are red
- Gather data
- Draw 10 cards

Hypothesis Testing

- Find the probability of the results under your

assumptions - Find the probability of drawing 10 red cards,

assuming that 50 of the 52 cards are red. - Probability of drawing 10 cards in a row is

highly unlikely if 50 of the 52 cards are red

(lt0.001).

Hypothesis Testing

- State your conclusion.
- Either we experienced a rare event, or one of our

assumptions is incorrect. - Since the probability of drawing 10 red cards is

small, we conclude that our assumptions are

probably incorrect. - We conclude that more than 50 of the cards are

red.

Hypothesis Testing ExampleIs There a Difference?

- Compare treatments or groups
- Psoriasis Example
- Some studies have suggested that psoriasis is

more common among heavy alcohol drinkers. - Case-control study of men age 19-50.
- Cases were men who had psoriasis.
- Controls were men who did not have psoriasis.
- All subjects completed questionnaires regarding

life style and alcohol consumption. - Is the mean alcohol intake for men with psoriasis

(cases) greater than men without psoriasis

(controls)? - Cases mean43, SD85.8, n142
- Controls mean21, SD34.2, n265
- Poikolainen et al Br Med J 1990 300780-783

Hypothesis TestingIs There a Difference?

- Null Hypothesis HO
- Often a statement of no treatment effect
- Example 1 The proportion of red cards is the

same as the proportion of black cards (50). - Example 2 There is no association between

alcohol intake and psoriasis. In other words,

the mean alcohol intake for men with psoriasis is

the same as the mean alcohol intake for men

without psoriasis.

Hypothesis TestingIs There a Difference?

- Alternative Hypothesis HA
- May be one-sided or two-sided
- Example 1
- One-sided The proportion of red cards is larger

than the proportion of black cards. - Two-sided The proportion of red cards is

different than the proportion of black cards. - Example 2
- One-sided Mean alcohol intake for cases (with

psoriasis) is larger than mean alcohol intake for

controls (without psoriasis) - Two-sided Mean alcohol intake for cases is

different than the mean alcohol intake for

controls

Hypothesis TestingConclusions

- The null hypothesis is assumed true until

evidence suggests otherwise. - 2 possible conclusions
- Reject the null hypothesis in favor of the

alternative. - Do not reject the null hypothesis.

Hypothesis Testing Errors

DECISION

Do not Reject HO

TRUTH

- Significance level a
- Probability of rejecting a true null hypothesis
- b
- Probability of not rejecting a false null

hypothesis - Power 1-b
- Probability of detecting a true difference

Reject HO

Type I Error (a)

Correct Decision

HO is True

HO is False

Correct Decision

Type II Error (b)

Hypothesis TestingSteps

- Assume the null hypothesis is true.
- Determine a test statistic based on the observed

data. - Using the test statistic, how likely is it that

we observe the outcome or something more extreme

if the null hypothesis is true? - If the test statistic is unlikely under the null

hypothesis, we reject the null hypothesis in

favor of the alternative hypothesis.

Hypothesis TestingP-value

- Measures how likely is it that we observe the

outcome or something more extreme, assuming the

null hypothesis is true. - Small p-value is evidence against the null

hypothesis and we reject the null hypothesis. - Large p-value suggests the data are likely if the

null hypothesis is true and we do not reject the

null hypothesis.

Hypothesis TestingP-value Method

- If p lt a, Reject the null in favor of the

alternative hypothesis. - If p gt a, Do Not Reject the null hypothesis.
- p lt .05 is generally considered statistically

significant. - Determining the p-value requires making

assumptions about the data.

Hypothesis TestingPsoriasis Example

- Ho There is no association between alcohol

intake and psoriasis. - Ha The mean alcohol intake is different for

cases and controls. - Using the test statistic, the p-value was 0.004.
- Conclusion Since the p-value is less than 0.05,

Reject Ho. - There is evidence that the mean alcohol intake is

higher for cases (mean43) than controls

(mean21).

Hypothesis TestingAntihypertensive Example

- Aim Compare two antihypertensive strategies for

lowering blood pressure - Double-blind, randomized study
- Enalapril Felodipine vs. Enalapril
- 6-week treatment period
- 217 patients
- Outcome of interest diastolic blood pressure
- Based on AJH, 199912691-696.

Hypothesis TestingAntihypertensive Example

- After 6 weeks of therapy, the average change in

DBP was - 10.6 mm Hg in the Enalapril Felodipine group

(n109, SD8.1) compared to - 7.4 mm Hg in the Enalapril group (n108, SD6.9)
- The authors used a hypothesis test to help

determine which therapy was more effective.

Hypothesis TestingAntihypertensive Example

- Statement from AJH
- The group randomized to 5 mg enalapril 5 mg

felodipine had a significantly greater reduction

in trough DBP after 6 weeks of blinded therapy

(10.6 mm Gh) than the group randomized to 10 mg

enalapril (7.4 mm Hg, Plt0.01). - What does Plt0.01 mean?
- Assuming that the 2 therapies are equally

effective, there is less than a 1 chance that we

would have observed treatment differences as

large or larger than what was observed.

Hypothesis Testing

- Parametric methods make assumptions about the

distribution of the observations. - Non-parametric methods do not make assumptions

about the distribution of the observations. - The distribution of the data and the design of

the study should be carefully considered when

choosing the statistical test to be used.

Comparing 2 Groups - Continuous Data Paired Data

- For each observation in the first group, there is

a corresponding observation in the second group. - Example Before and After
- Pairing eliminates some of the variability among

individuals, since measurements are made on the

same (or similar) subjects. - Paired groups are called dependent.

Comparing 2 Groups - Continuous DataPaired

t-test

- Two paired groups
- Sample size is large (30 or more pairs)

Normal Distribution

- Data follows a normal distribution if the

histogram is approximately symmetric and bell

shaped. - Described by two parameters
- mean (m)
- SD (s)

Normal Distribution

- Z-score measures how many SDs an observation is

away from the mean - Z(x-m)/s
- About 95 of the values fall within 2 SDs of the

mean

Comparing 2 Groups - Continuous Data Paired

t-test Example

- In 40 subjects, blood pressure was measured

before and after taking Captopril. - Outcome of interest change in blood pressure

after taking the drug - HO No association between Captopril and blood

pressure. - HA Mean blood pressure is lower after patients

take Captopril. - P-value lt 0.001.
- Reject HO in favor of HA. There is evidence that

mean blood pressure is lower after taking

Captopril. - Based on MacGregor et al., British Medical

Journal, Vol. 2

Comparing 2 Groups - Continuous Data Wilcoxon

Signed Ranks Test

- Two paired groups
- Sample size is small (less than 30 pairs).
- Wilcoxon Signed Ranks Test compares medians

rather than means. - Non-parametric test.

Comparing 2 Groups - Continuous Data Wilcoxon

Signed Ranks Test Example

- In 10 postcoronary patients, maximum oxygen

uptake was measured before and after a 6 month

exercise program. - Outcome of interest change in oxygen uptake

after a 6 month exercise program

Comparing 2 Groups - Continuous Data Wilcoxon

Signed Ranks Test Example

- HO There is no association between exercise and

oxygen uptake. - HA Median oxygen uptake is higher after

exercise program. - p-value .09.
- Do not reject HO. There is not enough evidence

to conclude that oxygen uptake is higher after

the exercise program.

Comparing 2 Groups - Continuous Data Independent

Samples t-test

- Two independent groups
- Sample size is large (30 or more in each group).

Comparing 2 Groups - Continuous Data Independent

Samples t-test Example

- 30 women with pregnancy-induced hypertension are

given low-dose aspirin - 42 women with pregnancy-induced hypertension

given a placebo - Outcome of interest blood pressure
- Based on Schiff, E et al., Obstetrics and

Gynecology, Vol 76, Nov 1990, 742-744.

Comparing 2 Groups - Continuous Data Independent

Samples t-test Example

- HO No association between low-dose aspirin and

blood pressure. - HA Mean blood pressure is lower for the aspirin

group - P-value 0.15.
- Do not reject HO. There is not enough evidence

to conclude that the mean blood pressure is lower

for the aspirin group.

Comparing 2 Groups - Continuous Data Wilcoxon

Rank Sum Test

- Two independent groups
- Sample size is small (less than 30).
- Wilcoxon Rank Sum Test compares medians rather

than means - Nonparametric test

Comparing 2 Groups - Continuous Data Wilcoxon

Rank Sum Test Example

- 13 patients randomized to placebo
- 15 randomized to receive calcium supplements
- Outcome of interest blood pressure
- HO No association between calcium supplements

and blood pressure. - HA Median blood pressure in calcium supplement

group is different than placebo group. - P-value .79.
- Do not reject HO. There is not enough evidence

to conclude that median blood pressure for the

calcium group is different than the placebo

group. - Based on Lyle et al., JAMA, Vol 257, No 13.

Comparing 3 or more groups

- Chi-square Test for categorical data
- Analysis of Variance (ANOVA) for continuous data
- Common uses
- Compare an outcome for 3 or more treatments
- Compare a characteristic in 3 or more populations

Chi-Square Test

- Compare 2 or more groups
- Categorical data
- Example To study effectiveness of bicycle

helmets, individuals who were in an accident were

studied. - Outcome of interest Compare proportion of

persons suffering a head injury while wearing a

helmet to proportion of persons suffering a head

injury while not wearing helmet

Chi-Square Test2x2 Table

Wearing Helmet Wearing Helmet

Injury Yes No

Yes No 17 (12) 130 (88) 218 (34) 428 (66)

Total 147 646

- 12 (17/147) of those wearing a helmet had a head

injury - 34 (218/646) of those not wearing a helmet had a

head injury

Chi-Square Test

- Ho The proportion suffering a head injury is the

same for accident victims who wore helmets vs.

accident victims who did not wear helmets. - Ha The proportion suffering a head injury is

different for accident victims who wore helmets

vs. accident victims who did not wear helmets. - p-value lt 0.001
- Conclusion Reject Ho. The proportion of

individuals suffering head injuries was higher

for accident victims who did not wear helmets

(34) compared to those who did wear helmets

(12). - Among persons in an accident, wearing a helmet

appears to lower incidence of head injury.

ANOVA (Analysis of variance)

- Used to compare a continuous variable among three

or more groups - HO The group (or treatment) means are the same.
- HA At least one mean is different from the

others.

One-Way ANOVA

- One factor (characteristic) is being studied
- Example treatment group
- Placebo
- experimental treatment 1
- experimental treatment 2
- 3 or more independent groups
- The distribution for each group is not heavily

skewed. - Group variances or sample sizes are approximately

equal.

One-Way ANOVAExample

- Aim Compare microbiological growth under 3

different CO2 pressure levels. - Factor of interest 3 different CO2 pressure

levels - Outcome of interest average microbiological

growth in each treatment group - HO The mean microbiological growth for the 3

treatments (CO2 level) is the same - HA At least one of the means is different.
- p-value .001
- Reject HO in favor of HA. There is evidence that

mean growth is different for the three treatment

groups.

One-Way ANOVAExample

- Mean microbiological growth under 3 different CO2

pressure levels. - Group 1 mean 56.2
- Group 2 mean 22.5
- Group 3 mean 26.1

Bonferroni Comparisons

- Use when ANOVA yields a significant p-value.
- If we perform several t-tests to compare each

pair of means, the probability of a Type I error

is gt 0.05. - The Bonferroni method modifies the p-value to

account for multiple comparisons so that,

overall, the probability of making a Type I error

is 0.05.

Bonferroni Comparisons Example

- Is the mean for group 1 different from the mean

for group 2? - P.001
- Conclusion The mean for group 1 is different

from the mean for group 2. - Is the mean for group 1 different from the mean

for group 3? - P.02
- Conclusion The mean for group 1 is different

from the mean for group 3. - Is the mean for group 2 different from the mean

for group 3? - P.34
- Conclusion The mean for group 2 is different

from the mean for group 3. - Therefore, the difference in the 3 group means

can primarily be explained by the higher mean for

group 1 compared to groups 2 and 3.

Repeated Measures ANOVA

- Subjects are measured at more than one time point
- Since multiple measurements are taken for the

same subject over time, the observations are not

independent

Repeated Measures ANOVA Example

- 12 rabbits receive in random order 3 different

dose levels of a drug to increase blood pressure,

with a washout period between treatments. - Outcome of interest average blood pressure for

the three dose levels - HO Average blood pressure is the same for the 3

dose levels - HA At least one of the means is different.
- P0.01
- Reject HO. There is evidence of a difference in

mean blood pressure for the 3 dose levels.

Kruskal-Wallis ANOVA

- Nonparametric ANOVA
- Use when the distribution for one or more groups

is heavily skewed.

Linear Regression

- Is there a linear relation between 2 continuous

variables? If so, what line best fits the data? - Use the line to predict a value for a new

observation - Example Can we predict muscle based on a

womans age? - Explore relationship between 2 numerical

variables - Example What is the relation between muscle

mass and age?

Linear Correlation (r)Is There an Association?

- Measures linear relationship between 2 continuous

variables. - Interpreting r

Absolute Value Linear of r Relationship 0 -

.25 poor .25 - .50 fair .50 - .75 good .75

1.0 very good

Linear Correlation (r)Examples

r .55

r 0

r -.85

r .85

Linear Correlation (r)Examples

r 1

r -1

Linear RegressionLeast Squares Regression Line

- Estimate the best line to fit the data
- Y b0 b1X
- Y is the dependent variable
- Example Muscle mass
- X is the independent variable
- Example Age of woman
- b0 is the intercept
- b1 is the slope

Linear Regression Example

- Predict the muscle mass of a 60 year old woman
- 148 - 60 80

Linear Regression Example

- On average, what is the difference in muscle mass

for women who differ in age by 1 year? - b1 -1
- For women whose age differs by one year, we

expect the average muscle mass will be one unit

lower for the older women

Linear RegressionNotes

- Significant correlation does not necessarily

imply causation. - Do not use a line to predict new observations if

there is not significant linear correlation. - When predicting new observations, stay within the

domain of the sample data.

References

- Dawson-Saunders, B and Trapp RG (1994). Basic

and Clinical Biostatistics. Appleton and Lange.

Norwalk, CT. - Lane, DM. (2000). Hyperstat Online. On-line

text, www.statistics.com. - MacGregor GA, Markandu ND, Roulston JE and Jones

JC (1979). Essential Hypertension Effect of

an Oral Inhibitor of Angiotensin-Converting

Enzyme. British Medical Journal, Nov 3 Vol 2,

1106-9. - Neter, J., Wasserman W. and Kutner, MH. (1990).

Applied Linear Statistical Models. Irwin. Burr

Ridge, IL. - Pagano M and Gauvreau, K. (1993). Principles of

Biostatistics. Duxbury Press. Belmont, CA. - Schiff E, Barkai G, Ben-Baruch G and Mashiach S.

(1990). Low-Dose Aspirin Does Not Influence the

Clinical Course of Women with Mild

Pregnancy-Induced Hypertension. Obstetrics and

Gynecology, Vol 76, November, 742-744. - Swinscow, TDV. (1997). Statistics at Square

One. BMJ Publishing Group. On-line text,

www.statistics.com. - Triola MF (1998), Elementary Statistics.

Addison-Wesley. Reading, MS.