Comparing Groups

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Comparing Groups

• Parametric Non-Parametric Inference

Nadeem Shafique Butt Dept. of Social Preventive
Paediatirics King Edward Medical University,
Lahore
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Parametric Non-Parametric Inference
Normality Un-Equal Variances
Normality Equal Variances
Normality Un-Equal Variances
Normality Equal Variances
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Comparing One Group

• Kinds of Research Questions
• For the one-sample situation, the prime concern
  in research is examining a measure of central
  tendency (location) for the population of
  interest. The best-known measures of location are
  the mean and median. For a one-sample situation,
  we might want to know if the average waiting time
  in a doctor's office is greater than one hour, or
  if the average growth of roses is 4 inches or
  more with a certain fertilizer, or is annual
  return is 10.2 for the banks that exercised
  comprehensive planning.

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Comparing Two Groups

• Kinds of Research Questions
• One of the most common tasks in research is to
  compare two populations (groups). We might want
  to compare the income level of two regions, the
  nitrogen content of two lakes, or the
  effectiveness of two drugs.
• The first question that arises is what aspects
  (parameters) of the populations shall we compare.
  We might consider comparing the averages, the
  medians, the standard deviations, the
  distributional shapes (histogram), or maximum
  values. We base the comparison parameter on our
  particular problem.
• Perhaps the simplest comparison that we can make
  is between the means of the two populations.

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Comparing more than two Groups

• Kinds of Research Questions
• One of the most common tasks in research is to
  compare several populations (groups). We might
  want to compare the income level of three
  regions, the nitrogen content of four lakes, or
  the effectiveness of four drugs.
• The first question that arises concerns which
  aspects (parameters) of the populations we should
  compare. We might consider comparing the means,
  medians, standard deviations, distributional
  shapes (histograms), or maximum values. We base
  the comparison of parameter on our particular
  problem.
• Perhaps the simplest comparison that we can make
  is to compare means of several populations.

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One Sample t-test

• One Sample t-test is used to compare one group to
  a given standard on the basis of Arithmetic
  Average (Mean).

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The assumptions of the One-sample t-test

• The data are continuous.
• The data follow the Normal distribution.
• The sample is a simple random sample from the
  population.

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Hypotheses and Formulas
With
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Example

• A manufacturer of high-performance automobiles
  produces disc brakes that must measure 322
  millimeters in diameter. Quality control manager
  randomly selects 128 discs and measures their
  diameters.
• We can use One Sample T Test to determine
  whether or not the mean diameters of the brakes
  in sample significantly differ from 322
  millimeters.

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SPSS Analytic Procedure
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The Sign Test

• The sign test is perhaps the oldest of all the
  nonparametric procedures. This nonparametric test
  is based on the binomial distribution. It assumes
  two mutually exclusive outcomes, constant or
  stable probability of success or failure, and n
  independent trials
• The terminology, sign test, reinforces the point
  that the data are converted to a series of plus
  and minus signs. The test is based on the number
  of plus signs that occur. Zero differences are
  thrown out, and the sample size is reduced
  accordingly.

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The Assumptionsof the Sign Test

• The data are continuous
• The distribution of these data is symmetric.
• The measurement scale is at least interval.

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Hypotheses and Formulas
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Example

• A health scientist believes that median survival
  time after breast cancer is 50 months. To confirm
  this hypothesis he selects a random sample of
  1207 breast cancer patients from different cancer
  hospitals.
• We can use Sign Test to determine whether or not
  the median survival time of the patients is
  significantly different from 50 months.

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SPSS Analytic Procedure
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Paired Samples t-test

• Kinds of Research Questions
• In the paired case, we take two measurements on
  same individual at different times, or we have
  one measurement on each individual of a pair.
• Examples of the first case are two
  insurance-claim adjusters assessing the damage
  for the same 15 cases. Evaluation of the
  improvement in aerobic fitness for 15 subjects
  where measurements are made at the beginning of
  the fitness program and at the end of it.
• An example of the second paired situation is the
  testing of the effectiveness of two drugs, A and
  B, on 20 pairs of patients who have been matched
  on physiological and psychological variables. One
  patient in the pair receives drug A, and the
  other patient gets drug B.

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The assumptions of the paired-sample t-test

• The data are continuous.
• The data, i.e., the differences for the
  matched-pairs, follow a Normal distribution.
• The sample of pairs is a simple random sample
  from its population.

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Hypotheses and Formulas
With
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Example

• A researcher in behavioral medicine believes
  that stress often makes asthma symptoms worse for
  people who suffer from this respiratory disorder.
  Therefore, the researcher decides to study the
  effect of relaxation training on the severity of
  their symptoms.
• A sample of 5 patients is selected. During the
  week before treatment, the investigator records
  the severity of their symptoms by measuring how
  many doses of medication are needed for asthma
  attacks. Then the patients receive relaxation
  training. For the week following the training the
  researcher once again records the number of doses
  used by each patient.
• Data from Gravetter and Wallnau (4th Ed.) p.
  319.

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SPSS Analytic Procedure
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Wilcoxon Signed Rank test

• Wilcoxon Signed Rank test is used to test the
  median difference of zero in case of non normal
  populations.

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The assumptions of the two-sample t-test

• The differences are continuous.
• The distribution of these differences is
  symmetric.
• The differences are mutually independent.
• The differences all have the same median.
• The measurement scale is at least interval.

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Hypotheses and Formulas
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Example

• An educationist to wants see the effectiveness
  of new teaching method. For this She selected 600
  students and record their scores in a test of 150
  marks. The scores are recorded before and after
  the new teaching method.
• The Wilcoxon Signed Rank test can be used to
  test the effectiveness of new teaching method.

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SPSS Analytic Procedure
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Independent Samples t-testEqual Variances

• Independent sample t test is used to compare
  two groups on the basis of their averages.

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The assumptions of the two-sample t-test

• The data are continuous
• The data follow the Normal distribution.
• The variances of the two populations are equal
• The two samples are independent
• Both samples are simple random samples from their
  respective populations.

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Hypotheses and Formulas
With
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Example

• An analyst at a department store wants to
  evaluate a recent credit card promotion. To this
  end, 500 cardholders were randomly selected. Half
  received an ad promoting a reduced interest rate
  on purchases made over the next three months, and
  half received a standard seasonal ad.
• We can use Independent-Samples T Test to compare
  the spending of the two groups.

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SPSS Analytic Procedure
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Independent Samples t-testUnequal Variances

• Independent Samples t-test is use to compare two
  independent groups on the basis of average. This
  test does not require homogeneity of the
  variances.

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Hypotheses and Formulas
With
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Example

• A researcher wishes to compare the expenditure
  behavior of the students, one of the research
  question is to see the difference in expenditures
  by gender.

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SPSS Analytic Procedure
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Mann-Whitney Test

• Mann-Whitney Test is used to compare the two
  independent groups on the basis of medians. This
  test does not require the assumption of normality.

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Mann-Whitney U Test Assumptions

• The variable of interest is continuous. The
  measurement scale is at least ordinal.
• The probability distributions of the two
  populations are identical, except for location.
• The two samples are independent.
• Both samples are simple random samples from their
  respective populations.

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Hypotheses and Formulas
W is the sum of ranks of the smaller sample
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Example

• Data on birth weight of infants born to mothers
  with different levels of prenatal care. Two
  independent samples data for univariate analysis.
  Test data for Mann-Whitney U-Test, obtained from
  Howell, David D. Fundamental Statistics for the
  Behavioral Sciences 3rd Edition, p385.

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SPSS Analytic Procedure
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One-Way Analysis of VarianceEqual Variances

• One Way Analysis of Variance is used to compare
  more than two groups on the basis of their
  averages.

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One-Way Analysis of Variance Assumptions

• The data are continuous.
• The data follow the Normal distribution, each
  group is normally distributed.
• The variances of the populations are equal.
• The groups are independent.
• Each group is a simple random sample from its
  population.

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Hypotheses and Formulas
MSG is the Mean Square of Group and MSE is the
Mean Square Error
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Example

• This is a hypothetical data file that concerns
  the popularity of a TV channel. Using a
  prototype, the marketing team has collected focus
  group data. One of the question of interest is to
  see the difference in popularity of the TV
  channel in different age groups.
• This hypothesis can be tested using One Way
  ANOVA.

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SPSS Analytic Procedure
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One-Way Analysis of VarianceUnequal Variances

• Welch ANOVA is used to compare more than two
  groups on the basis of averages. This test doest
  not require the homogeneity of variances.

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Welch Analysis of Variance Assumptions

• The data are continuous
• The data follow the Normal distribution, each
  group is normally distributed.
• The groups are independent.
• Each group is a simple random sample from its
  population.

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Hypotheses and Formulas
With
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Example

• A sales manager evaluates two new training
  courses.
• Sixty employees, divided into three groups, all
  receive standard training. In addition, group 2
  receives technical training, and group 3 receives
  a hands-on tutorial. Each employee was tested at
  the end of the training course and their score
  recorded.

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SPSS Analytic Procedure
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Kruskal-Wallis Test

• Kruskal-Wallis H-test is used to compare more
  than two groups on the basis of their medians.

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Kruskal-Wallis Test Assumptions

• The variable of interest is continuous, the
  measurement scale is at least ordinal.
• The probability distributions of the populations
  are identical, except for location.
• The groups are independent.
• All groups are simple random samples from their
  respective populations.

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Hypotheses and Formulas
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Example

• A health scientist wishes to compare the
  survival experiences after breast cancer with
  different Pathological Tumor Size (Categories).
• We can use Kruskal-Wallis H-Test to determine
  whether or not the median survival time of the
  patients is significantly differ in different
  pathological tumor size.

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SPSS Analytic Procedure