Inferential Data Analysis

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Chapter 15

• Inferential Data Analysis

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Inferential Statistics

• Inferential statistics are a very crucial part
  of scientific research in that these techniques
  are used to test hypotheses.

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Uses for Inferential Statistics

• Statistics for determining differences between
  experimental and control groups in experimental
  research
• Statistics are also used in descriptive research
  when comparisons are made between different
  groups

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• These statistics enable the researcher to
  evaluate the effects of an independent variable
  on a dependent variable.

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Sampling Error

• Remember when we talked about the sampling error?
• Parameters characteristics of a population
• Statistics characteristics of a sample

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• Differences between a sample statistic and a
  population parameter because the sample is not
  perfectly representative of the population.
• So, maybe the differences among the sample means
  may be a real difference, or it may be due to
  sampling error.

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• Researcher needs a standard procedure to follow
  in making such a decision.

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Hypothesis-testing procedure
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Hypothesis testing

• Hypothesis testing or significance testing is
  used to determine whether what you observed in
  the sample provides enough evidence to believe
  that there is a difference in the population.
• In other words the sample difference is said to
  be statistically different or not statistically
  different.

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Hypothesis Testing

• The Research Hypothesis is transformed into a
  Statistical or Null Hypothesis.
• This is done so that statistical tests can be
  employed that will determine whether the findings
  are statistically significant or can be
  attributed to chance.
• The results of the statistical test will enable
  the researcher to accept or reject the null
  hypothesis.

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More Hypothesis Testing

• The purpose of the statistical test is to
  evaluate the null hypothesis at a specified level
  of probability
• For instance, testing the difference in the mean
  values between 2 groups at the .05 level means

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• Do the values of the dependent variable differ
  significantly (plt.05) so that these differences
  would not be attributable to chance occurrence
  more than 5 times in 100?

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Level of Significance

• Rejecting the null hypothesis at the .05 alpha
  (?) level suggests a 95 probability that the
  differences between the two variables is real and
  not the result of chance.

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• Type I Error rejecting a null hypothesis when it
  is really true.
• Probability of making a type I error is equal to
  ?
• Type II Error acceptance or not rejecting a null
  hypothesis when it is false.
• Probability of making a type II error is equal
  to ß.

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Hypothesis Testing Procedures

• State the hypothesis (H0), and then you select
  the probability level (alpha).
• The researcher sets a significance level to
  indicate the maximum risks she is willing to take
  of making an error when concluding that there is
  a difference attributable to the research
  situation and not to chance.
• Commonly used ? 0.05 or 0.01.

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• Next, you should decide if you are going to use a
  one-tailed or a two-tailed test.
• In a one-tail test, the 5 area of rejection is
  either at the upper end or the lower end.

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To reject the null, the tail used for the
rejection region should cover the extreme
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The t or z scores that are rejected are ones in
the red region or positive values.
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• In a two-tail test, the 5 area of rejection is
  split between the upper and lower tails of the
  curve null hypothesis is nondirectional.

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• The next thing is that you need to determine if a
  parametric test or a nonparametric test should be
  used.
• Most common mistakes doing a parametric test
  when the data are not normally distributed.

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Parametric Nonparametric Tests

• Parametric
• Assume the data are normally distributed.
• That the dependent variables are continuous and
  measured on an interval or ratio scale.
• Parametric tests are powerful tools make maximum
  use of the information available in the data
  (e.g. mean, standard deviation).

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Normal curve

• The normal curve is a statistical model that is
  used to visualize data, interpret distributions
  of scores, and make predictions and probability
  statements.
• Mean, median, and mode are identical, and makes
  up the vertical midpoint.
• 95 of the area is between 2 SD.

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Normal Curve
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• Nonparametric
• Make no assumptions about the population under
  investigation.
• Can be used with nominal or ordinal data.
• Can be used for very small samples, or large
  samples which do not fit parametric test
  assumptions.

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• Major drawback with these tests is that they are
  less powerful than their parametric analogs.
• Increased chance of a Type II error less
  sensitive to small differences and less able to
  detect that such differences might be
  statistically significant.

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• Check
• Data is normally distributed?
• Population variances of the groups are
  approximately equal?
• That the dependent variables are continuous and
  measured on an absolute, interval, or ratio
  scale?
• Large or small sample size?

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• If youre not sure there are a number of
  statistical tests to see if your data is
  parametric or nonparametric. These tests are
  called tests for normality or goodness of fit
  test.
• Eg Kuipers goodness of fit
• Watsons goodness of fit Liliefors test for
  normality
• Kolmogorov-Smirnov goodness of fit

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To reiterate yesterdays class

• Hypothesis-testing procedure
• State the hypothesis
• Select the probability level (?)
• Decide if the data are normal or not normal.
• Consult the statistical table.

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Parametric Tests
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t-tests

• Characteristics of t-tests
• requires interval or ratio level scores
• used to compare two mean scores
• easy to compute
• pretty good small sample statistic

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Types of t-test

• One-Group t-test
• t-test between a constant and a sample mean
• Independent Groups t-test
• compares mean scores on two independent samples

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Total and saturated fat intake by year of study.
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Total and saturated fat intake as affected by
major and year of study.
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• Dependent Groups or paired (Correlated) t-test
• compares two mean scores from a repeated measures
  or matched pairs design
• most common situation is for comparison of
  pretest with posttest scores from the same sample

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Analysis of Variance (ANOVA)

• Analysis of Variance compares two or more means
  to determine if there are statistically
  significant differences among them.
• Compares the amount of variation between the
  groups with the variation within the groups.
• If ANOVA determines that differences exist, data
  are then further tested with post hoc test

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• Examples of post hoc tests
• Duncans multiple range test
• Student-Newman-Keuls test (SNK)
• Tukeys Honestly Significant Difference (HSD)
• Scheffes test
• These tests determine how the individual means
  differ.

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One-way ANOVA

• Extension of independent groups t-test, but may
  be used for evaluating differences among 2 or
  more groups.

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Repeated Measures ANOVA

• Each subject is measured on 2 or more occasions
• a.k.a within subjects design.
• Example three Ca dosage, measure 5 times each

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Random Blocks ANOVA

• An extension of the matched pairs t-test when
  there are three or more groups or the same as the
  matched pairs t-test when there are two groups.
• Participants similar in terms of a variable are
  placed together in a block, then randomly
  assigned to treatment groups another source of
  variation that you want to eliminate

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• Example three doses of Ca given to subjects
  from ethnic groups. Assign subjects from each
  ethnic group to each treatment.

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Factorial ANOVA

• This is an extension of the one-way ANOVA for
  testing the effects of 2 or more independent
  variables as well as interaction effects.
• Two-way ANOVA (e.g., 3 X 2 ANOVA)
• Three-way ANOVA (e.g., 3 X 3 X 2 ANOVA)
• Example 3 doses of Ca (1st variable) and sex of
  the subjects (2nd variable).

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Multivariate Analysis of Variance (MANOVA)

• MANOVA determines whether differences exist
  between two or more treatments considering
  several variables at the same time.
• Replaces repeating ANOVA on each variable
• Determines whether treatments differ on balance
  considering all variables at once (e.g. blood
  chemistry).

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Nonparametric Tests
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The Chi-Square Test

• Used to test the distribution of observations in
  a sample among classes
• Determines whether a single sample is different
  from a hypothetical distribution (e.g., normal
  distribution), or whether two or more
  distributions differ from each other.
• Example incidence of obesity (on a 1-5 scale -
  counts) in men and women

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Single Sample Chi-Square

• a.k.a one-way chi-square or goodness of fit
  chi-square
• Used to test the hypothesis that the collected
  data (observed scores) fits an expected
  distribution

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Independent Groups Chi-Square

• a.k.a. two-way chi-square or contingency table
  chi-square
• Used to test if there is a significant
  relationship (association) between two nominally
  scaled variables
• In this test we are comparing two or more
  patterns of frequencies to see if they are
  independent from each other

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The Mann-Whitney U-Test

• Nonparametric equivalent of a t-test requires no
  assumptions about the nature of the data.

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The Wilcoxon Matched-Pair Signed Rank Test

• An nonparametric test equivalent to the t-test
  for paired samples
• Applied when each sample from one treatment is
  matched with a sample from the other treatment
  (by subject age, for example)

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The Kruskal-Wallis Test

• The nonparametric equivalent of a one-way or
  paired Analysis of Variance

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Finally
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• Conduct the statistical test

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• Accept or reject the null hypothesis.
• If the p value for the value of the statistical
  test is less than the alpha level, the null
  hypothesis is rejected.
• If the opposite is true, the null hypothesis is
  accepted.

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Hypothesis Testing Errors

• Remember, the researcher sets a significance
  level to indicate the maximum risks she is
  willing to take of making an error when
  concluding that there is a difference
  attributable to the research situation and not to
  chance.
• Commonly used ? 0.05 or 0.01.

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• Hypothesis testing decisions are made without
  direct knowledge of the true circumstance in the
  population. As a result, the researchers
  decision may or may not be correct.

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Possible Decisions in Statistical Significance
Testing