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UT SW 388R7 - Testing Multivariate Assumptions

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Testing Multivariate AssumptionsTesting Multivariate Assumptions - 21. Evaluating the Normality of Metric VariablesRequesting Statistics to Test NormalityRequesting the Plot to Test NormalityOutput for the Statistical Tests of NormalityThe Histogram for Delivery Speed (X1)The Normality Plot for Delivery Speed (X1)The Histogram for Price Level (X2)The Normality Plot for Price Level (X2)Transformations to Induce NormalityComputing the Square Root Transformation for Price LevelRequest the Normality Analysis for the Transformed Price Level VariableThe K-S Lilliefors Test for the Transformed Price Level VariableThe Histogram for the Transformed Price Level VariableThe Normality Plot for the Transformed Price Level VariableThe Histogram for Price Flexibility (X3)The Normality Plot for Price Flexibility (X3)Computing the Square Root Transformation for Price FlexibilityComputing the Logarithmic Transformation for Price FlexibilityComputing the Inverse Transformation for Price FlexibilityRequest the explore command for the three transformed variablesThe K-S Lilliefors tests for the transformed variables2. Evaluating Homogeneity of Variance for Non-metric VariablesRequesting a One-way ANOVARequest the Levene Homogeneity of Variance TestThe Tests of Homogeneity of VariancesCompute the Transformed Variables for 'Manufacturer Image' (x4)Request the Levene Test for the Transformed Manufacturer Image VariablesLevene Test Results for the Transformed Manufacturer Image VariablesCompute the Transformed Variables for 'Product Quality' (x7)Request the Levene Test for the Transformed Product Quality VariablesResults of the Levene Test for the Transformed Product Quality Variables3. Evaluate Linearity and Homoscedasticity of Metric Variables with ScatterplotsRequesting the Scatterplot MatrixSpecify the Variables to Include in the Scatterplot MatrixAdd Fit Lines to the Scatterplot MatrixRequesting the Fit LinesChanging the Thickness of the Fit LinesChanging the Color of the Fit LinesThe Final Scatterplot MatrixSlide 1 Testing Multivariate AssumptionsThe multivariate statistical techniques which we will cover in this class require one or more the following assumptions about the data: normality of the metric variables, homoscedastic relationships between the dependent variable and the metric and nonmetric independent variables, linear relationships between the metric variables, and absence of correlated prediction errors.Multivariate analysis requires that the assumptions be tested twice: first, for the separate variables as we are preparing to do the analysis, and second, for the multivariate model variate, which acts collectively for the variables in the analysis and thus must meet the same assumptions as individual variables.! In this section, we will examine the tests that we normally perform prior to computing the multivariate statistic.! Since the pattern of prediction errors cannot be examined without computing the multivariate statistic, we will defer that discussion until we examine each of the specific techniques.If the data fails to meet the assumptions required by the analysis, we can attempt to correct the problem with a transformation of the variable.! There are two classes of transformations that we attempt: for violations of normality and homoscedasticity, we transform the individual metric variable to a inverse, logarithmic, or squared form; for violations of linearity, we either do a power transformation, e.g. raise the data to a squared or square root power, or we add an additional polynomial variable that contains a power term.! Testing Multivariate AssumptionsSlide 2 Testing Multivariate Assumptions - 2Transforming variables is a trial and error process.! We do the transformation and then see if it has corrected the problem with the data.! It is not usually possible to be certain in advance that the transformation will correct the problem; sometimes it only reduces the degree of the violation.! Even when the transformation might decrease the violation of the assumption, we might opt not to include it in the analysis because of the increased complexity it adds to the interpretation and discussion of the results.It often happens that one transformation solves multiple problems.! For example, skewed variables can produce violations of normality and homoscedasticity.! No matter which test of assumptions identified the violation, our only remedy is a transformation of the metric variable to reduce the skewness.Testing Multivariate AssumptionsSlide 3 1. Evaluating the Normality of Metric VariablesDetermining whether or not the distribution of values for a metric variable complies with the definition of a normal curve is tested with histograms, normality plots, and statistical tests.The histogram shows us the relative frequency of different ranges of values for the variable.! If the variable is normally distributed, we expect the greatest frequency of values to occur in the center of the distribution, with decreasing frequency for values away from the center.! In addition, a normally distributed variable will be symmetric, showing the same proportion of cases in the left and right tails of the distribution.In a normality plot in SPSS, the actual distribution of cases is plotted in red against the distribution of cases that would be expected if the variable is normally distributed, plotted as a green line on the chart.! Our conclusion about normality is based on the convergence or divergence between the plot of red points and the green line.There are two statistical tests for normality:! the Kolmogorov-Smirnov statistic with the Lilliefors correction factor for variables that have 50 cases or more, and the Shapiro-Wilk's test for variables that have fewer than 50 cases.! SPSS will compute the test which is appropriate to the sample size.! The statistical test is regarded as sensitive to violations of normality, especially for a large sample, so we should examine the histogram and normality plot for confirmation of a distribution problem.The statistical test for normality is a test of the null hypothesis that the distribution is normal.! The desirable outcome is a significance value for the statistic more than 0.05 so that we fail to reject the null hypothesis. If we fail to reject the null hypothesis, we conclude that the variable is normally distributed and meets the normality assumption.! If the significance value of the normality test statistic is smaller than 0.05, we reject the null


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