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**Chapter 14 Inferential Data Analysis**

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**Inferential Statistics**

Techniques that allow us to study samples and then make generalizations about the population. 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 used in descriptive research when comparisons are made between different groups These statistics enable the researcher to evaluate the effects of an independent variable on a dependent variable

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Sampling Error Differences between a sample statistic and a population parameter because the sample is not perfectly representative of the population

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Hypothesis Testing The purpose of the statistical test is to evaluate the null hypothesis (H0) at a specified level of significance (e.g., p < .05) In other words, do the treatment effects differ significantly so that these differences would be attributable to chance occurrence less than 5 times in 100?

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**Hypothesis Testing Procedures**

State the hypothesis (H0) Select the probability level (alpha) Determine the value needed for significance Calculate the test statistic Accept or reject H0

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**Statistical Significance**

A statement in the research literature that the statistical test was significant indicates that the value of the calculated statistic warranted rejection of the null hypothesis For a difference question, this suggests a real difference and not one due to sampling error

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**Parametric Statistics**

Techniques which require basic assumptions about the data, for example: normality of distribution homogeneity of variance requirement of interval or ratio data Most prevalent in HHP Many statistical techniques are considered robust to violations of the assumptions, meaning that the outcome of the statistical test should still be considered valid

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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 Independent Groups t-test**

t-test between a sample and population mean Independent Groups t-test compares mean scores on two independent samples Dependent Groups (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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**Hypothesis Testing Errors**

Hypothesis testing decisions are made without direct knowledge of the true circumstance in the population. As a result, the researcher’s decision may or may not be correct Type I Error Type II Error

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Type I Error . . . is made when the researcher rejects the null hypothesis when in fact the null hypothesis is true probability of committing Type I error is equal to the significance (alpha) level set by the researcher thus, the smaller the alpha level the lower the chance of committing a Type I error

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Type II Error . . . occurs when the researcher accepts the null hypothesis, when in fact it should have been rejected probability is equal to beta (B) which is influenced by several factors inversely related to alpha level increasing sample size will reduce B Statistical Power – the probability of rejecting a false null hypothesis Power = 1 – beta Decreasing probability of making a Type II error increases statistical power

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**Hypothesis Truth Table**

NULL HYPOTHESIS TRUE FALSE CORRECT DECISION TYPE II ERROR ACCEPT DECISION TYPE I ERROR CORRECT DECISION REJECT

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**ANOVA - Analysis of Variance**

A commonly used family of statistical tests that may be considered a logical extension of the t-test requires interval or ratio level scores used for comparing 2 or more mean scores maintains designated alpha level as compared to experimentwise inflation of alpha level with multiple t-tests may also test more than 1 independent variable as well as interaction effect

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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**

Extension of dependent groups t-test, where each subject is measured on 2 or more occasions a.k.a “within subjects design” Test of sphericity assumption is recommended

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Random Blocks ANOVA This is 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 and then randomly assigned to treatment groups

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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)

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**Assumptions of Statistical Tests**

Parametric tests are based on a variety of assumptions, such as Interval or ratio level scores Random sampling of participants Scores are normally distributed N = 30 considered minimum by some Homogeneity of variance Groups are independent of each other Others Researchers should try to satisfy assumptions underlying the statistical test being used

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**Improving the Probability of Meeting Assumptions**

Utilize a sample that is truly representative of the population of interest Utilize large sample sizes Utilize comparison groups that have about the same number of participants

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**Two-Group Comparison Tests**

a.k.a. Multiple Comparison or Post Hoc Tests The various ANOVA tests are often referred to as “omnibus” tests because they are used to determine if the means are different but they do not specify the location of the difference if the null hypothesis is rejected, meaning that there is a difference among the mean scores, then the researcher needs to perform additional tests in order to determine which means (groups) are actually different

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Common Post Hoc Tests Multiple comparison (post hoc) tests are used to make specific comparisons following a significant finding from ANOVA in order to determine the location of the difference Duncan Tukey Bonferroni Scheffe Note that post hoc tests are only necessary if there are more than two levels of the independent variable

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**Analysis of Covariance**

ANOVA ANOVA design which statistically adjusts the difference among group means to allow for the fact that the groups differ on some other variable frequently used to adjust for inequality of groups at the start of a research study

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**Nonparametric Statistics**

Considered assumption free statistics Appropriate for nominal and ordinal data or in situations where very small sample sizes (n < 10) would probably not yield a normal distribution of scores Less statistical power than parametric statistics

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Chi Square A nonparametric test used with nominally scaled data which are common with survey research The statistic is used when the researcher is interested in the number of responses, objects, or people that fall in two or more categories

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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 i.e. are the observed frequencies and expected frequencies for a questionnaire item in agreement with each other?

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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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**Overview of Multivariate Tests**

Univariate statistic – used in situations where each participant contributed one score to the data analysis, or in the case of a repeated measures design, one score per cell Multivariate statistic – used in situations where each participant contributes multiple scores

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**Example Multivariate Tests**

MANOVA Canonical correlation Discriminant analysis Factor analysis

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**Multiple Analysis of Variance**

MANOVA Analogous to ANOVA except that there are multiple dependent variables Represents a type of multivariate test

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**Prediction and Regression Analysis**

Correlational technique Simple prediction Predicting an unknown score (Y) based on a single predictor variable (X) Y’ = bX + c Multiple prediction Involves more than one predictor variable Y’ = b1X1 + b2X2 + c

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**Multiple Regression/Prediction**

a.k.a multiple correlation Determines the relationship between one dependent variable and 2 or more predictor variables Used to predict performance on one variable from another Y’ = b1X1 + b2X2 + c Standard error of prediction is an index of accuracy of the prediction

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Statistical Power The probability that the statistical test will correctly reject a false null hypothesis . . . it is effectively the probability of finding significance, that the experimental treatment actually does have an effect a researcher would like to have a high level of power

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**Statistical Power alpha = probability of a Type I error**

rejecting a true null hypothesis this is your significance level beta = probability of a Type II error failing to reject a false null hypothesis Statistical power = 1 - beta

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**Factors Affecting Power**

Alpha level Sample size Effect size One-tailed or two-tailed test

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Alpha level Reducing the alpha level (moving from .05 to .01) will reduce the power of a statistical test. This makes it harder to reject the null hypothesis

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Sample size In general, the larger the sample size the greater the power. This is because the standard error of the mean decreases as the sample size increases

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**One-tailed versus two-tailed tests**

It is easier to reject the null hypothesis using a one-tailed test than a two-tailed test because the critical region is larger

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Effect size This is an indication of the size of the treatment effect, its meaningfulness With a large effect size, it will be easy to detect differences and statistical power will be high But, if the treatment effect is small, it will be difficult to detect differences and power will be low

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Effect Size Numerous authors have indicated the need to estimate the magnitude of differences between groups as well as to report the significance of the effects One way to describe the strength of a treatment effect, or meaningfulness of the findings, is the computation of “effect size” (ES) ES = M1 - M2 SD Note: SD represents the standard deviation of the control group or the pooled standard deviation if there is no control group

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**Effect Size Interpretation of ES by Cohen (1988)**

0.2 represents a small ES 0.5 represents a moderate ES 0.8 represents a large ES Researchers using experimental designs are advised to provide post hoc estimates of ES for any significant findings as a way to evaluate the meaningfulness

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A Priori Procedures Calculate the power for each of the statistical procedures to be applied requires three indices - alpha, sample size, effect size Estimate the sample size needed to detect a certain effect (ES) given a specific alpha and power may require an estimation of ES from previous published studies or from a pilot study

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