Chapter 9 Comparing Two Groups

1
Chapter 9Comparing Two Groups

• Learn .
• How to Compare Two Groups On a Categorical or
  Quantitative Outcome Using Confidence Intervals
  and Significance Tests

2
Bivariate Analyses

• The outcome variable is the response variable
• The binary variable that specifies the groups is
  the explanatory variable

3
Bivariate Analyses

• Statistical methods analyze how the outcome on
  the response variable depends on or is explained
  by the value of the explanatory variable

4
Independent Samples

• The observations in one sample are independent of
  those in the other sample
• Example Randomized experiments that randomly
  allocate subjects to two treatments
• Example An observational study that separates
  subjects into groups according to their value for
  an explanatory variable

5
Dependent Samples

• Data are matched pairs each subject in one
  sample is matched with a subject in the other
  sample
• Example set of married couples, the men being
  in one sample and the women in the other.
• Example Each subject is observed at two times,
  so the two samples have the same people

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Section 9.1

• Categorical Response How Can We Compare Two
  Proportions?

7
Categorical Response Variable

• Inferences compare groups in terms of their
  population proportions in a particular category
• We can compare the groups by the difference in
  their population proportions
• (p1 p2)

8
Example Aspirin, the Wonder Drug

• Recent Titles of Newspaper Articles
• Aspirin cuts deaths after heart attack
• Aspirin could lower risk of ovarian cancer
• New study finds a daily aspirin lowers the risk
  of colon cancer
• Aspirin may lower the risk of Hodgkins

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Example Aspirin, the Wonder Drug

• The Physicians Health Study Research Group at
  Harvard Medical School
• Five year randomized study
• Does regular aspirin intake reduce deaths from
  heart disease?

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Example Aspirin, the Wonder Drug

• Experiment
• Subjects were 22,071 male physicians
• Every other day, study participants took either
  an aspirin or a placebo
• The physicians were randomly assigned to the
  aspirin or to the placebo group
• The study was double-blind the physicians did
  not know which pill they were taking, nor did
  those who evaluated the results

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Example Aspirin, the Wonder Drug

• Results displayed in a contingency table

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Example Aspirin, the Wonder Drug

• What is the response variable?
• What are the groups to compare?

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Example Aspirin, the Wonder Drug

• The response variable is whether the subject had
  a heart attack, with categories yes or no
• The groups to compare are
• Group 1 Physicians who took a placebo
• Group 2 Physicians who took aspirin

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Example Aspirin, the Wonder Drug

• Estimate the difference between the two
  population parameters of interest

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Example Aspirin, the Wonder Drug

• p1 the proportion of the population who would
  have a heart attack if they participated in this
  experiment and took the placebo
• p2 the proportion of the population who would
  have a heart attack if they participated in this
  experiment and took the aspirin

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Example Aspirin, the Wonder Drug
Sample Statistics
17
Example Aspirin, the Wonder Drug

• To make an inference about the difference of
  population proportions, (p1 p2), we need to
  learn about the variability of the sampling
  distribution of

18
Standard Error for Comparing Two Proportions

• The difference, , is obtained from
  sample data
• It will vary from sample to sample
• This variation is the standard error of the
  sampling distribution of

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Confidence Interval for the Difference between
Two Population Proportions

• The z-score depends on the confidence level
• This method requires
• Independent random samples for the two groups
• Large enough sample sizes so that there are at
  least 10 successes and at least 10 failures
  in each group

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Confidence Interval Comparing Heart Attack Rates
for Aspirin and Placebo

• 95 CI

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Confidence Interval Comparing Heart Attack Rates
for Aspirin and Placebo

• Since both endpoints of the confidence interval
  (0.005, 0.011) for (p1- p2) are positive, we
  infer that (p1- p2) is positive
• Conclusion The population proportion of heart
  attacks is larger when subjects take the placebo
  than when they take aspirin

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Confidence Interval Comparing Heart Attack Rates
for Aspirin and Placebo

• The population difference (0.005, 0.011) is small
• Even though it is a small difference, it may be
  important in public health terms
• For example, a decrease of 0.01 over a 5 year
  period in the proportion of people suffering
  heart attacks would mean 2 million fewer people
  having heart attacks

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Confidence Interval Comparing Heart Attack Rates
for Aspirin and Placebo

• The study used male doctors in the U.S
• The inference applies to the U.S. population of
  male doctors
• Before concluding that aspirin benefits a larger
  population, wed want to see results of studies
  with more diverse groups

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Interpreting a Confidence Interval for a
Difference of Proportions

• Check whether 0 falls in the CI
• If so, it is plausible that the population
  proportions are equal
• If all values in the CI for (p1- p2) are
  positive, you can infer that (p1- p2) gt0
• If all values in the CI for (p1- p2) are
  negative, you can infer that (p1- p2) lt0
• Which group is labeled 1 and which is labeled
  2 is arbitrary

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Interpreting a Confidence Interval for a
Difference of Proportions

• The magnitude of values in the confidence
  interval tells you how large any true difference
  is
• If all values in the confidence interval are near
  0, the true difference may be relatively small in
  practical terms

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Significance Tests Comparing Population
Proportions

• 1. Assumptions
• Categorical response variable for two groups
• Independent random samples

27
Significance Tests Comparing Population
Proportions

• Assumptions (continued)
• Significance tests comparing proportions use the
  sample size guideline from confidence intervals
  Each sample should have at least about 10
  successes and 10 failures
• Twosided tests are robust against violations of
  this condition
• At least 5 successes and 5 failures is
  adequate

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Significance Tests Comparing Population
Proportions

• 2. Hypotheses
• The null hypothesis is the hypothesis of no
  difference or no effect
• H0 (p1- p2) 0
• Under the presumption that p1 p2, we create a
  pooled estimate of the common value of p1and p2
• This pooled estimate is

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Significance Tests Comparing Population
Proportions

• 2. Hypotheses (continued)
• Ha (p1- p2) ? 0 (two-sided test)
• Ha (p1- p2) lt 0 (one-sided test)
• Ha (p1- p2) gt 0 (one-sided test)

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Significance Tests Comparing Population
Proportions

• 3. The test statistic is

31
Significance Tests Comparing Population
Proportions

• 4. P-value Probability obtained from the
  standard normal table
• 5. Conclusion Smaller P-values give stronger
  evidence against H0 and supporting Ha

32
Example Is TV Watching Associated with
Aggressive Behavior?

• Various studies have examined a link between TV
  violence and aggressive behavior by those who
  watch a lot of TV
• A study sampled 707 families in two counties in
  New York state and made follow-up observations
  over 17 years
• The data shows levels of TV watching along with
  incidents of aggressive acts

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Example Is TV Watching Associated with
Aggressive Behavior?
34
Example Is TV Watching Associated with
Aggressive Behavior?

• Test the Hypotheses
• H0 (p1- p2) 0
• Ha (p1- p2) ? 0
• Using a significance level of 0.05
• Group 1 less than 1 hr. of TV per day
• Group 2 at least 1 hr. of TV per day

35
Example Is TV Watching Associated with
Aggressive Behavior?
36
Example Is TV Watching Associated with
Aggressive Behavior?

• Conclusion Since the P-value is less than 0.05,
  we reject H0
• We conclude that the population proportions of
  aggressive acts differ for the two groups
• The sample values suggest that the population
  proportion is higher for the higher level of TV
  watching

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Section 9.2

• Quantitative Response How Can We Compare Two
  Means?

38
Comparing Means

• We can compare two groups on a quantitative
  response variable by comparing their means

39
Example Teenagers Hooked on Nicotine

• A 30-month study
• Evaluated the degree of addiction that teenagers
  form to nicotine
• 332 students who had used nicotine were evaluated
• The response variable was constructed using a
  questionnaire called the Hooked on Nicotine
  Checklist (HONC)

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Example Teenagers Hooked on Nicotine

• The HONC score is the total number of questions
  to which a student answered yes during the
  study
• The higher the score, the more hooked on nicotine
  a student is judged to be

41
Example Teenagers Hooked on Nicotine

• The study considered explanatory variables, such
  as gender, that might be associated with the HONC
  score

42
Example Teenagers Hooked on Nicotine

• How can we compare the sample HONC scores for
  females and males?
• We estimate (µ1 - µ2) by (x1 - x2)
• 2.8 1.6 1.2
• On average, females answered yes to about one
  more question on the HONC scale than males did

43
Example Teenagers Hooked on Nicotine

• To make an inference about the difference between
  population means, (µ1 µ2), we need to learn
  about the variability of the sampling
  distribution of

44
Standard Error for Comparing Two Means

• The difference, , is obtained from
  sample data. It will vary from sample to sample.
• This variation is the standard error of the
  sampling distribution of

45
Confidence Interval for the Difference between
Two Population Means

• A 95 CI
• Software provides the t-score with right-tail
  probability of 0.025

46
Confidence Interval for the Difference between
Two Population Means

• This method assumes
• Independent random samples from the two groups
• An approximately normal population distribution
  for each group
• this is mainly important for small sample sizes,
  and even then the method is robust to violations
  of this assumption

47
Example Nicotine How Much More Addicted Are
Smokers than Ex-Smokers?

• Data as summarized by HONC scores for the two
  groups
• Smokers x1 5.9, s1 3.3, n1 75
• Ex-smokersx2 1.0, s2 2.3, n2 257

48
Example Nicotine How Much More Addicted Are
Smokers than Ex-Smokers?

• Were the sample data for the two groups
  approximately normal?
• Most likely not for Group 2 (based on the sample
  statistics) x2 1.0, s2 2.3)
• Since the sample sizes are large, this lack of
  normality is not a problem

49
Example Nicotine How Much More Addicted Are
Smokers than Ex-Smokers?

• 95 CI for (µ1- µ2)
• We can infer that the population mean for the
  smokers is between 4.1 higher and 5.7 higher than
  for the ex-smokers

50
How Can We Interpret a Confidence Interval for a
Difference of Means?

• Check whether 0 falls in the interval
• When it does, 0 is a plausible value for (µ1
  µ2), meaning that it is possible that µ1 µ2
• A confidence interval for (µ1 µ2) that contains
  only positive numbers suggests that (µ1 µ2) is
  positive
• We then infer that µ1 is larger than µ2

51
How Can We Interpret a Confidence Interval for a
Difference of Means?

• A confidence interval for (µ1 µ2) that contains
  only negative numbers suggests that (µ1 µ2) is
  negative
• We then infer that µ1 is smaller than µ2
• Which group is labeled 1 and which is labeled
  2 is arbitrary

52
Significance Tests Comparing Population Means

• 1. Assumptions
• Quantitative response variable for two groups
• Independent random samples

53
Significance Tests Comparing Population Means

• Assumptions (continued)
• Approximately normal population distributions for
  each group
• This is mainly important for small sample sizes,
  and even then the two-sided test is robust to
  violations of this assumption

54
Significance Tests Comparing Population Means

• 2. Hypotheses
• The null hypothesis is the hypothesis of no
  difference or no effect
• H0 (µ1- µ2) 0

55
Significance Tests Comparing Population
Proportions

• 2. Hypotheses (continued)
• The alternative hypothesis
• Ha (µ1- µ2) ? 0 (two-sided test)
• Ha (µ1- µ2) lt 0 (one-sided test)
• Ha (µ1- µ2) gt 0 (one-sided test)

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Significance Tests Comparing Population Means

• 3. The test statistic is

57
Significance Tests Comparing Population Means

• 4. P-value Probability obtained from the
  standard normal table
• 5. Conclusion Smaller P-values give stronger
  evidence against H0 and supporting Ha

58
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• Experiment
• 64 college students
• 32 were randomly assigned to the cell phone group
• 32 to the control group

59
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• Experiment (continued)
• Students used a machine that simulated driving
  situations
• At irregular periods a target flashed red or
  green
• Participants were instructed to press a brake
  button as soon as possible when they detected a
  red light

60
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• For each subject, the experiment analyzed their
  mean response time over all the trials
• Averaged over all trials and subjects, the mean
  response time for the cell-phone group was 585.2
  milliseconds
• The mean response time for the control group was
  533.7 milliseconds

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Example Does Cell Phone Use While Driving
Impair Reaction Times?

• Data

62
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• Test the hypotheses
• H0 (µ1- µ2) 0
• vs.
• Ha (µ1- µ2) ? 0
• using a significance level of 0.05

63
Example Does Cell Phone Use While Driving
Impair Reaction Times?
64
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• Conclusion
• The P-value is less than 0.05, so we can reject
  H0
• There is enough evidence to conclude that the
  population mean response times differ between the
  cell phone and control groups
• The sample means suggest that the population mean
  is higher for the cell phone group

65
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• What do the box plots tell us?
• There is an extreme outlier for the cell phone
  group
• It is a good idea to make sure the results of the
  analysis arent affected too strongly by that
  single observation
• Delete the extreme outlier and redo the analysis
• In this example, the t-statistic changes only
  slightly

66
Example Does Cell Phone Use While Driving
Impair Reaction Times?

• Insight
• In practice, you should not delete outliers from
  a data set without sufficient cause (i.e., if it
  seems the observation was incorrectly recorded)
• It is however, a good idea to check for
  sensitivity of an analysis to an outlier
• If the results change much, it means that the
  inference including the outlier is on shaky ground

67
How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week.
Gender Sample Size Mean St. Dev.
Women 6764 32.6 18.2
Men 4252 18.1 12.9

• What is a point estimate of µ1- µ2?
• 18.2 12.9
• 32.6 18.1
• 6764 - 4252
• 32.6/18.2 18.1/12.9

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How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week.
Gender Sample Size Mean St. Dev.
Women 6764 32.6 18.2
Men 4252 18.1 12.9

• What is the standard error for comparing the
  means?
• 5.3
• .076
• .297
• .088

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How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week. How much more time do women spend on housework than men? Data is Hours per Week.
Gender Sample Size Mean St. Dev.
Women 6764 32.6 18.2
Men 4252 18.1 12.9

• What factor causes the standard error to be
  small compared to the sample standard deviations
  for the two groups?
• sample means
• sample standard deviations
• sample sizes
• genders

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Section 9.3

• Other Ways of Comparing Means and Comparing
  Proportions

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Alternative Method for Comparing Means

• An alternative t- method can be used when, under
  the null hypothesis, it is reasonable to expect
  the variability as well as the mean to be the
  same
• This method requires the assumption that the
  population standard deviations be equal

72
The Pooled Standard Deviation

• This alternative method estimates the common
  value s of s1 and s1 by

73
Comparing Population Means, Assuming Equal
Population Standard Deviations

• Using the pooled standard deviation estimate, a
  95 CI for (µ1 - µ2) is
• This method has df n1 n2- 2

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Comparing Population Means, Assuming Equal
Population Standard Deviations

• The test statistic for H0 µ1µ2 is
• This method has df n1 n2- 2

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Comparing Population Means, Assuming Equal
Population Standard Deviations

• These methods assume
• Independent random samples from the two groups
• An approximately normal population distribution
  for each group
• This is mainly important for small sample sizes,
  and even then, the CI and the two-sided test are
  usually robust to violations of this assumption
• s1s2