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Inferential Data Analysis

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Title: Inferential Data Analysis


1
Chapter 15
  • Inferential Data Analysis

2
Inferential Statistics
  • Inferential statistics are a very crucial part
    of scientific research in that these techniques
    are used to test hypotheses.

3
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…

4
  • These statistics enable the researcher to
    evaluate the effects of an independent variable
    on a dependent variable.

5
Sampling Error
  • Remember when we talked about the sampling error?
  • Parameters characteristics of a population
  • Statistics characteristics of a sample

6
  • 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.

7
  • Researcher needs a standard procedure to follow
    in making such a decision.

8
Hypothesis-testing procedure
9
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.

10
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.

11
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

12
  • 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?

13
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.

14
  • 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.

15
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.

16
  • 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.

17
To reject the null, the tail used for the
rejection region should cover the extreme
18
The t or z scores that are rejected are ones in
the red region or positive values.
19
  • 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.

20
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21
  • 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.

22
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).

23
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.

24
Normal Curve
25
  • 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.

26
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27
  • 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.

28
  • 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?

29
  • 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

30
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31
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.

32
Parametric Tests
33
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

34
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

35
Total and saturated fat intake by year of study.
36
Total and saturated fat intake as affected by
major and year of study.
37
  • 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

38
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

39
  • 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.

40
One-way ANOVA
  • Extension of independent groups t-test, but may
    be used for evaluating differences among 2 or
    more groups.

41
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

42
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

43
  • Example three doses of Ca given to subjects
    from ethnic groups. Assign subjects from each
    ethnic group to each treatment.

44
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).

45
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).

46
Nonparametric Tests
47
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

48
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

49
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

50
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51
The Mann-Whitney U-Test
  • Nonparametric equivalent of a t-test requires no
    assumptions about the nature of the data.

52
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)

53
The Kruskal-Wallis Test
  • The nonparametric equivalent of a one-way or
    paired Analysis of Variance

54
Finally…
55
  • Conduct the statistical test…

56
  • 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.

57
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.

58
  • 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.

59
Possible Decisions in Statistical Significance
Testing
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