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SPSS Workshop Research Support Center Chongming Yang

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Causal Inference If A, then B, under condition C If A, 95% Probability B, under condition C

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Student T Test (William S. Gossett’s pen name = student) Assumptions – Small Sample – Normally Distributed t distributions: t = [ x - μ ] / [ s / sqrt( n ) ] df = degrees of freedom=number of independent observations

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Type of T Tests One sample – test against a specific (population) mean Two independent samples – compare means of two independent samples that represent two populations Paired – compare means of repeated samples

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One Sample T Test Conceputally convert sample mean to t score and examine if t falls within acceptable region of distribution

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Two Independent Samples

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Paired Observation Samples d = difference value between first and second observations

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Multiple Group Issues Groups A B C comparisons – AB AC BC – Joint Probability that one differs from another –.95*.95*.95 =.91

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Analysis of Variance (ANOVA) Completely randomized groups Compare group variances to infer group mean difference Sources of Total Variance – Within Groups – Between Groups F distribution – SSB = between groups sum squares – SSW = within groups sum squares

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Fisher-Snedecor Distribution

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F Test

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Issues of ANOVA Indicates some group difference Does not reveal which two groups differ Needs other tests to identify specific group difference – Hypothetical comparisons Contrast – No Hypothetical comparisons Post Hoc ANOVA has been replaced by multiple regressions, which can also be replaced by General Linear Modeling (GLM)

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Multiple Linear Regression

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Assumptions of Linear Regression Y and X have linear relations Y is continuous or interval & unbounded expected or mean of = 0 = normally distributed not correlated with predictors Predictors should not be highly correlated No measurement error in all variables

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Least Squares Solution

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Model Comparisons Complete Model: Reduced Model: Test F = Ms drop / MSE – MS = mean square – MSE = mean square error

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Variable Selection Select significant from a pool of predictors Stepwise undesirable, see Forward Backward (preferable)

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R = Race(1=white, 2=Black, 3=Hispanic, 4=Others) R d1 d2 d Include all dummy variables in the model, even if not every one is significant.

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Interaction Create a product term X 2 X 3 Include X 2 and X 3 even effects are not significant Interpret interaction effect: X 2 effect depends on the level of X 3.

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Plotting Interaction Write out model with main and interaction effects, Use standardized coefficient Plug in some plausible numbers of interacting variables and calculate y Use one X for X dimension and Y value for the Y dimension See examples

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Diagnostic Linear relation of predicted and observed (plotting Collinearity Outliers Normality of residuals (save residual as new variable)

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Repeated Measures (MANOVA, GLM) Measure(s) repeated over time Change in individual cases (within)? Group differences (between, categorical x)? Covariates effects (continuous x)? Interaction between within and between variables?

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Assumptions Normality Sphericity: Variances are equal across groups so that Total sum of squares can be partitioned more precisely into – Within subjects – Between subjects – Error

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Model

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F Test of Effects F = MS between / Ms within (simple repeated) F = Ms treatment / Ms error (with treatment) F = Ms within / Ms interaction (with interaction)

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Four Types Sum-Squares Type I balanced design Type II adjusting for other effects Type III no empty cell unbalanced design Type VI empty cells

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Exercise epeated_Measures/default.htm epeated_Measures/default.htm Copy data to spss syntax window, select and run Run Repeated measures GLM

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