Introduction to Introduction to Biostatistics IIBiostatistics II
Jane L. Meza, Ph.D.Jane L. Meza, Ph.D.
October 24, 2005October 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 (<0.001).
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 Example:Is 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: mean=43, SD=85.8, n=142 Controls: mean=21, SD=34.2, n=265Poikolainen et al Br Med J 1990; 300:780-783
Hypothesis Testing:Is 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 Testing:Is 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 Testing:Conclusions
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
Significance level: Probability of rejecting a true null
hypothesis
Probability of not rejecting a false null hypothesis
Power: 1- Probability of detecting a true difference
Type I Error ()
Type II Error ()
Correct Decision
Correct Decision
DECISION
Reject HO
Do notReject HO
TRUTH
HO is False
HO is True
Hypothesis Testing:Steps 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 Testing:P-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 Testing:P-value Method If p < , Reject the null in favor
of the alternative hypothesis.
If p >= , Do Not Reject the null hypothesis.
p < .05 is generally considered statistically significant.
Determining the p-value requires making assumptions about the data.
Hypothesis Testing:Psoriasis 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 (mean=43) than controls (mean=21).
Hypothesis Testing:Antihypertensive 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, 1999;12:691-696.
Hypothesis Testing:Antihypertensive Example After 6 weeks of therapy, the
average change in DBP was:
10.6 mm Hg in the Enalapril + Felodipine group (n=109, SD=8.1) compared to
7.4 mm Hg in the Enalapril group (n=108, SD=6.9)
The authors used a hypothesis test to help determine which therapy was more effective.
Hypothesis Testing:Antihypertensive 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, P<0.01).”
What does P<0.01 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 Data
Paired 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 () SD ()
Normal Distribution
Z-score measures how many SDs an observation is away from the mean
Z=(x-)/ 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 < 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
Difference in max. oxygen uptake ml/(kg)(min)
5.00.0-5.0-10.0-15.0-20.0
Difference in Maximum Oxygen Uptake
Before and After Exercise Program
Fre
qu
en
cy
5
4
3
2
1
0
Std. Dev = 8.10
Mean = -5.2
N = 10.00
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
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
Wearing Helmet
Injury Yes No
Yes
No
17 (12%)
130 (88%)
218 (34%)
428 (66%)
Total 147 646
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 < 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 > 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.
P=0.01 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 woman’s age? Explore relationship between 2
numerical variables Example: What is the relation between
muscle mass and age?
Y = 148 - X
X = AGE (years)
8070605040
Y =
Me
asu
re o
f M
usc
le M
ass
120
110
100
90
80
70
60
Linear Correlation (r)Is There an Association? Measures linear relationship between 2
continuous variables.
Interpreting r :
AbsoluteValue Linearof r Relationship0 - .25 poor.25 - .50 fair.50 - .75 good.75 – 1.0 very good
Linear Correlation (r)Examples
r = .55r = 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
Y = 148 - X
X = AGE (years)
8070605040
Y =
Me
asu
re o
f M
usc
le M
ass
120
110
100
90
80
70
60
Predict the muscle mass of a 60 year old woman 148 - 60 = 80
Linear Regression ExampleY = 148 - X
X = AGE (years)
8070605040
Y =
Me
asu
re o
f M
usc
le M
ass
120
110
100
90
80
70
60
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.