Introduction to Biostatistics II

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

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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

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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?

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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?

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Experiment

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

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Hypothesis Testing

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

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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).

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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.

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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

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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.

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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

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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.

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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)
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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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Comparing 2 Groups - Continuous DataPaired
t-test

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

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Normal Distribution

• Data follows a normal distribution if the
  histogram is approximately symmetric and bell
  shaped.
• Described by two parameters
• mean (m)
• SD (s)

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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

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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

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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.

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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

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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.

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Comparing 2 Groups - Continuous Data Independent
Samples t-test

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

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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.

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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.

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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

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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.

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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

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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

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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

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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

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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.

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Kruskal-Wallis ANOVA

• Nonparametric ANOVA
• Use when the distribution for one or more groups
  is heavily skewed.

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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?

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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
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Linear Correlation (r)Examples
r .55
r 0
r -.85
r .85
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Linear Correlation (r)Examples
r 1
r -1
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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

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Linear Regression Example

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

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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

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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.

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References

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  JC (1979). Essential Hypertension Effect of
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