Simple Statistical Designs One Dependent Variable

Is your Dependent Variable (DV) continuous? YES NO Is your Independent Variable (IV) continuous? Correlation or Linear Regression YES Do you have only 2 treatments? NO Logistic Regression Chi Square NO T-testANOVA If I have one Dependent Variable, which statistical test do I use?

Chi Square

Chi Square (χ 2 )  Non-parametric: no parameters estimated from the sample  Chi Square is a distribution with one parameter: degrees of freedom (df).  Positively skewed but skew decreases with df.  Mean is df  Goodness-of-fit and Independence Tests

Chi-Square Goodness of Fit Test  How well do observed proportions or frequencies fit theoretically expected proportions or frequencies?  Example: Was test performance better than chance? χ 2 =Σ (Observed – Expected) 2 df = # groups -1 Expected ObservedExpected Correct6250 Incorrect3850

Chi Square Test for Independence  Is distribution of one variable contingent on another variable?  Contingency Table  df = (#Rows -1)(#Columns-1)  Example: H o : depression & gender are independent H 1 : depression and gender are not independent MaleFemaleTotal Depressed10(15)20(15)30 Not Depressed 40(35)30(35)70 Total

Chi Square Test for Independence Same χ 2 formula except expected frequencies are derived from the row and column totals: cell proportion X Total = (30/100)(50/100)(100) χ 2 = (10-15) 2 + (20-15) 2 + (40-35) 2 + (30-15) 2 = Critical χ 2 with 1 df = 3.84 at p=.05 Reject H o : depression and gender are NOT independent MaleFemaleTotal Depressed10(15)20(15)30 Not Depressed 40(35)30(35)70 Total

Assumptions of Chi Square  Independence of observations  Categories are mutually exclusive  Sampling distribution in each cell is normal  Violated if expected frequencies are very low ( 20.  Fisher’s Exact Test can correct for violations of these assumptions in 2x2 designs.

Correlation and Regression

Recall the Bivariate Distribution Recall the Bivariate Distribution r = -.17 p=.09

Interpretation of r  Slope of best fitting straight regression line when variables are standardized  measure of the strength of the relationship between 2 variables  r 2 measures proportion of variability in one measure that can be explained by the other  1-r 2 measures the proportion of unexplained variability.

Correlation Coefficients Coefficient Variable 1 Type Variable 2 Type Pearson r continuouscontinuous Point Biserial continuousdichotomy Phi Coefficient dichotomydichotomy Biserialcontinuous Artificial dichotomy Tetrachoric Spearman’s Rho ranksranks

Simple Regression  Prediction: What is the best prediction of variable X?  Regress Y on X (i.e. regress outcome on predictor)  CorrelationRegression.html CorrelationRegression.html

The fit of a straight line  The straight line is a summary of a bivariate distribution  Y = a + bx + ε  DV = intercept + slope(IV) + error  Least Squares Fit: minimize error by minimizing sum of squared deviations: Σ(Actual Y - Predicted Y) 2  Regression lines ALWAYS pass through the mean of X and mean of Y

b  Slope: the magnitude of change in Y for a 1 unit change in X  Beta= b = r(SD y / SD x )  Because of this relationship: Z y = r Z x  Standardized beta: if X and Y are converted to Z scores, this would be the beta – not interpretable as slope.

Residuals  The error in the estimate of the regression line  Mean is always 0  Residual plots are very informative – tell you how well your line fits the data  Linear Regression Applet Linear Regression Applet Linear Regression Applet

Assumptions & Violations Linear Regression Applet Linear Regression Applet Linear Regression Applet  Homoscedasticity: uniform variance across whole bivariate distribution.  Bivariate outlier: not outlier on either X or Y  Influential Outliers: ones that move the regression line  Y is Independent and Normally distributed at all points along line (residuals are normally distributed)  Omission of important variables  Non-linear relationship of X and Y  Mismatched distributions (i.e. neg skew and pos skew – but you already corrected those with transformations, right?)  Group membership (i.e. neg r within groups, pos r across groups)

Logistic Regression  Continuous predictor(s) but DV is now dichotomous.  Predicts probability of dichotomous outcome (i.e. pass/fail, recover/relapse)  Not least squares but maximum likelihood estimate  Fewer assumptions than multiple regression  “Reverse” of ANOVA  Similar to Discriminant Function Analysis that predicts nominal-scaled DVs of > 2 categories

T-test  Similar to Z but with estimates instead of actual population parameters mean1 – mean2 pooled within-group SD  One- or two-tailed, use one-tailed if you can justify through hypothesis - more power  Effect size is Cohen’s d

One Sample t-test Compare mean of one variable to a specific value (i.e. Is IQ in your sample different from national norm?) Sample mean –

Independent Sample t-test  Are 2 groups significantly different from each other?  Assumes independence of groups, normality in both populations, and equal variances (although T is robust against violations of normality).  Pooled variance = mean of variances (or weighted by df if variances are unequal)  If N’s unequal, use Welch t-test

Dependent Samples t-test (aka Paired Samples t-test)  Dependent Samples:  Same subjects, same variables  Same subjects, different variables  Related subjects, same variables (i.e. mom and child)  More powerful: pooled variance (denominator) is smaller  But fewer df, higher critical t

Univariate (aka One-Way) ANOVA AnalysisofVariance  2 or more levels of a factor  ANOVA tests H o that means of each level are equal  Significant F only indicates that the means are not equal.

F  F statistic = t 2 = Between Group Variance = signal Within Group Variance noise Robust against violations of normality unless n is small Robust against violations of homogeneity of variances unless n’s are unequal If n’s are unequal, use Welch F’ or Brown-Forsythe F*

Effect size  Large F does NOT equal large effect  Eta Squared (η 2 ): Sum-of-Squares between Sum-of-squares Total Sum-of-squares Total Variance proportion estimate Positively biased – OVERestimates true effect  Omega squared (ω 2 ) adjusts for within factor variability and is better estimate

Family-wise error  F is a non-directional, omnibus test and provides no info about specific comparisons between factors. In fact, a non-significant omnibus F does not mean that there are not significant differences between specific means.  However, you can’t just run a separate test for each comparison – each independent test has an error rate (α).  Family-wise error rate = 1 – (1- α) c, where c = # comparisons  Example: 3 comparisons with α=.05 1 – (1-.05) 3 =.143

Contrasts  A linear combination of contrast coefficients (weights) on the means of each level of the factor Control Drug 1 Drug 2 mean10205 To contrast the Control group against the Drug 1 group, the contrast would look like this: Contrast = 1(Control) + (-1)(Drug 1) + 0(Drug 2)

Unplanned (Post-hoc) Contrasts  Risk of Family-wise error  Correct with:  Bonferoni inequailty: multiply α by # comparisons  Tukey’s Honest Significant Difference (HSD): minimum difference between means necessary for significance  Scheffe test: critical F’ = (#groups-1)(F) ultraconservative

Planned Contrasts  Polynomial: linear, quadratic, cubic, etc. pattern of means across levels of the factor  Orthogonal: sum of contrast coefficients (weights) equals 0.  Non-orthogonal: sum of contrast coefficients does not equal 0

Polynomial Contrasts (aka Trend Analysis)  Special case of orthogonal contrasts but IV must be ordered (e.g. time, age, drug, dosage) LinearQuadraticCubicQuartic

Orthogonal Contrasts  Deviation : Compares the mean of each level (except one) to the mean of all of the levels (grand mean). Levels of the factor can be in any order. Control Drug 1 Drug 2 Grand Mean

Orthogonal Contrasts Simple: Compares the mean of each level to the mean of a specified level. This type of contrast is useful when there is a control group. You can choose the first or last category as the reference. Control Drug 1 Drug 2 Grand Mean

Orthogonal Contrasts Helmert : Compares the mean of each level of the factor (except the last) to the mean of subsequent levels combined. Control Drug 1 Drug 2 Grand Mean

Orthogonal Contrasts Difference : Compares the mean of each level (except the first) to the mean of previous levels. (aka reverse Helmert contrasts.) Control Drug 1 Drug 2 Grand Mean

Orthogonal Contrasts Repeated : Compares the mean of each level (except the last) to the mean of the subsequent level. Control Drug 1 Drug 2 Grand Mean

Non-orthogonal Contrasts  Not used often  Dunn’s test (Bonforoni t): controls for family-wise error rate by multiplying α by the number of comparisons.  Dunnett’s test: use t-test but critical t values come from a different table (Dunnett’s) that restricts family-wise error.