Paired Samples and Blocks

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Chapter 25

• Paired Samples and Blocks

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Paired Data

• Paired data arise in a number of ways.
• Compare subjects with themselves before and after
  treatment
• Blocking pairs arise from an experiment
• Matching pairs arise from an observational study

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Matched Pairs t-test

• Treat the differences as if they were the data,
  ignoring the original columns of data.
• Since we have one column of values, we will use a
  simple one-sample
• t-test.
• A matched pair t-test is just a one-sample t-test
  for the means of these pairwise differences.

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Assumptions and Conditions

• Paired data assumption
• Data must be paired the samples cant be
  independent.
• Justify your claim that the data are paired.
• Independence assumption
• Randomization What we want to know usually
  focuses our attention on where the randomness
  should be.

• Independence assumption
• 10 condition When the inference is about a
  population from which the paired individuals are
  drawn, we must be sure that we have sampled no
  more than 10 of that population.
• Normal population assumption
• Nearly Normal condition Check with a histogram
  or Normal probability plot of the differences.

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A Paired t-test for the Mean Differences Between
Two Groups

• When the conditions are met, test whether the
  paired differences differ significantly from
  zero.
• We test the hypothesis
• where the ds are the pairwise differences and
• ?O is almost always zero.
• We use the statistic

• n is the number of pairs and
• When the conditions are met and the null
  hypothesis is true, this statistic follows a
  Students t-model on n 1 degrees of freedom, so
  we can use that model to obtain a P-value.

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Health Dept. Workers and Mileage

• Hypothesis
• HO The mileage driven by each health dept worker
  during a four-day work week is the same as his
  mileage for a five-day work week. The mean
  difference is zero.
• HA The mean difference is different from zero.

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Health Dept. Workers and Mileage

• Paired data assumption The data are paired
  because hey are measurements on the same
  individuals before and after a change in work
  schedule.
• Independence assumption The behavior of any
  individual is independent of the behavior of
  others, so the differences are mutually
  independent.

• RandomizationThe measured values are the sums of
  individual trips, each of which experienced
  random events that arose while driving.
• 10 condition Our inference is about driving
  amounts, not about the workers, so we dont need
  to check this condition.

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Health Dept. Workers and Mileage

• Nearly Normal Condition The histogram of the
  differences is unimodal and symmetric.

• Under these conditions the sampling distribution
  of the differences can be modeled by a Students
  t-model with (n 1) 10 degrees of freedom.
• We will use a paired
• t-test.

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Health Dept. Workers and Mileage

• Find from the data
• We know

• STAT TESTS T-Test

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Health Dept. Workers and Mileage

• Interpretation
• With a P-value this small, we can reject the null
  hypothesis.
• We conclude that the change in the work week did
  lead to a change in driving mileage.
• We should look at the confidence interval. If the
  difference in mileage proves to be large in a
  practical sense, then we might recommend a change
  in schedule for the rest of the department.

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Confidence Intervals for Matched Pairs

• Paired t-interval
• When the conditions are met, find the confidence
  interval for the mean of the paired differences.
• Since the standard error of the mean
• difference is
• The confidence interval is
• The critical value t depends on the particular
  confidence level C that is specified and on the
  degrees of freedom (n 1) which is based on the
  number of pairs, n.

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Husbands and Wives

• We wish to find an interval that is likely with
  95 confidence to contain the true mean
  difference in ages of husbands and wives.
• Histogram (1st 16 pairs)

• Conditions
• Paired data assumption The data are paired
  because they are members of married couples.
• Randomization These couples were randomly
  selected.
• 10 condition The sample is less than 10 of the
  population of married couples in Britain.
• Nearly Normal condition The histogram of the
  differences is unimodal and symmetric.

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Husbands and Wives

• Under these conditions, the sampling distribution
  of the differences can be modeled by a Students
  t-model with (n 1) 169 degrees of freedom.
• We will find a paired
• t-interval.

• We know
• STAT TEST TInterval

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Husbands and Wives

• Interpretation
• We are 95 confident that in married couples in
  Britain, the husband is, on average, between 1.6
  and 2.8 years older than his wife.

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Blocking

• Pairing removes the extra variation and allows us
  to focus on the individual differences.
• In experiments, we block to separate the
  variability between the the experimental units
  from the variability in the response.

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Caution!!

• Dont use a two-sample t-test for paired data.
• Dont use a paired-t method when the samples
  arent paired.
• Dont forget outliers.
• Dont look for the difference in side-by-side
  boxplots. A scatterplot of the two variables can
  sometimes be helpful.