Introduction to Biostatistics

Jane L. Meza, Ph.D.

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

• Is it possible that we could draw 10 red cards in a row from a standard deck of cards?

• Is it very likely that we could draw 10 red cards in a row 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 in a row, 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 in a row is so 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. (Poikolainen et al Br Med J 1990; 300:780-783)

– 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=265

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.

• If p ≥ , Do Not Reject the null.

• 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 .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 Hg) 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”• Example: Subjects age/sex

matched

• 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 < .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 = .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 were randomized to placebo

• 15 take 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

YesNo

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 < .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 > 5%.

• The Bonferroni method modifies the p-value to account for multiple comparisons so that, overall, the probability of making a Type I error is 5%.

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=.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 Relationship 0 - .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 =

Mea

sure

of M

uscl

e 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 =

Mea

sure

of M

uscl

e 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.