Reasoning in Psychology Using Statistics 2018

Don’t forget quiz 8 due this Friday Annoucements

Measures of Error in Regression The linear equation isn’t the whole thing Also need a measure of error Y = X(.5) + (2.0) + error Y X 1 2 3 4 5 6 Y = X(.5) + (2.0) + error Same line, but different relationships (strength difference) Y X 1 2 3 4 5 6 Measures of Error in Regression

Measures of Error in Regression The linear equation isn’t the whole thing Also need a measure of error Three common measures of error r2 (r-squared) R-squared (r2) represents the percent variance in Y accounted for by X Sum of the squared residuals = SSresidual= SSerror Compute the difference between the predicted values and the observed values (“residuals”) Square the differences Add up the squared differences Standard error of estimate Y X 1 2 3 4 5 6 Measures of Error in Regression Statistics by Jim: Standard Error of the Regression vs. R-squared

Measures of Error in Regression Sum of the squared residuals = SSresidual = SSerror Y X 1 2 3 4 5 6 6.2 1.6 5.3 3.45 These are all points on the prediction line X Y 6 6 6.2 = (0.92)(6)+0.688 1 2 1.6 = (0.92)(1)+0.688 5 6 5.3 = (0.92)(5)+0.688 3 4 3.45 = (0.92)(3)+0.688 3 2 3.45 = (0.92)(3)+0.688 mean 3.6 4.0 Measures of Error in Regression

Measures of Error in Regression Sum of the squared residuals = SSresidual = SSerror residuals X Y These are deviations between the points on the prediction line and the actual observed values (in the Y direction) Y X 1 2 3 4 5 6 6 6 6.2 6 - 6.2 = -0.20 1 2 1.6 2 - 1.6 = 0.40 5 6 5.3 6 - 5.3 = 0.70 3 4 3.45 4 - 3.45 = 0.55 3 2 3.45 -1.45 2 - 3.45 = mean 3.6 4.0 Quick check 0.00 Measures of Error in Regression

Measures of Error in Regression Sum of the squared residuals = SSresidual = SSerror X Y 6 6 6.2 -0.20 0.04 1 2 1.6 0.40 0.16 5 6 5.3 0.70 0.49 3 4 3.45 0.55 0.30 3 2 3.45 -1.45 2.10 mean 3.6 4.0 0.00 3.09 SSERROR Measures of Error in Regression

Measures of Error in Regression Sum of the squared residuals = SSresidual = SSerror 4.0 0.0 16.0 SSY X Y 6 6 6.2 -0.20 0.04 1 2 1.6 0.40 0.16 5 6 5.3 0.70 0.49 3 4 3.45 0.55 0.30 3 2 3.45 -1.45 2.10 mean 3.6 4.0 0.00 3.09 SSERROR Don’t get these confused with each other Measures of Error in Regression

Measures of Error in Regression Standard error of the estimate represents the average deviation from the line Y X 1 2 3 4 5 6 df = n - 2 Measures of Error in Regression

Measures of Error in Regression SPSS Regression output gives you a lot of stuff r2 percent variance in Y accounted for by X Standard error of the estimate the average deviation from the line SSresiduals or SSerror Measures of Error in Regression

Chi-Square Test for Independence A manufacturer of watches takes a sample of 200 people. Each person is classified by age and watch type preference (digital vs. analog). Young (under 30) Old (over 30) The question: Is there a relationship between age and watch preference? Chi-Square Test for Independence

Chi-Square Test for Independence A manufacturer of watches takes a sample of 200 people. Each person is classified by age and watch type preference (digital vs. analog). Young (under 30) Old (over 30) The question: Is there a relationship between age and watch preference? Chi-Square Test for Independence

Decision tree Chi-square test of independence (χ2 lower-case chi ) Describing the relationship between two categorical variables or Young Old or Decision tree

Chi-Squared Test for Independence A manufacturer of watches takes a sample of 200 people. Each person is classified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference? Chi-Squared Test for Independence

Chi-Squared Test for Independence A manufacturer of watches takes a sample of 200 people. Each person is classified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference? Step 1: State the hypotheses and select an alpha level H0: Preference is independent of age (“no relationship”) HA: Preference is related to age (“there is a relationship”) We’ll set α = 0.05 Observed scores Chi-Squared Test for Independence

Chi-Squared Test for Independence Step 2: Compute your degrees of freedom & get critical value df = (#Columns - 1) * (#Rows - 1) = (3-1) * (2-1) = 2 Go to Chi-square statistic table and find the critical value The critical chi-squared value is 5.99 For this example, with df = 2, and α = 0.05 Chi-Squared Test for Independence

Chi-Squared Test for Independence As df gets larger, need larger X2 value for significance. Number of cells get larger. X2 α = .05 5.99 7.81 11.07 14.07 Chi-Squared Test for Independence

Chi-Squared Test for Independence Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies Observed scores Chi-Squared Test for Independence

Chi-Squared Test for Independence Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies Observed scores Spot check: make sure the row totals and column totals add up to the same thing Chi-Squared Test for Independence

Chi-Squared Test for Independence Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies (in each cell) Observed scores Expected scores 70 56 14 30 24 6 Under 30 Over 30 Digital Analog Undecided Chi-Squared Test for Independence

Chi-Squared Test for Independence Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies (in each cell) Observed scores Expected scores 70 56 14 “expected frequencies” - if the null hypothesis is correct, then these are the frequencies that you would expect 30 24 6 Under 30 Over 30 Digital Analog Undecided Chi-Squared Test for Independence

Chi-Squared Test for Independence Step 4: compute the χ2 Find the residuals (fo - fe) for each cell Chi-Squared Test for Independence

Computing the Chi-square Step 4: compute the χ2 Find the residuals (fo - fe) for each cell Computing the Chi-square

Computing the Chi-square Step 4: compute the χ2 Find the residuals (fo - fe) for each cell Square these differences Computing the Chi-square

Computing the Chi-square Step 4: compute the χ2 Find the residuals (fo - fe) for each cell Square these differences Divide the squared differences by fe Computing the Chi-square

Computing the Chi-square Step 4: compute the χ2 Find the residuals (fo - fe) for each cell Square these differences Divide the squared differences by fe Sum the results Computing the Chi-square

Chi-Squared, the final step A manufacturer of watches takes a sample of 200 people. Each person is classified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference? Step 5: Compare this computed statistic (38.09) against the critical value (5.99) and make a decision about your hypotheses here we reject the H0 and conclude that there is a relationship between age and watch preference Chi-Squared, the final step

Chi square as a statistical test each cell = observed difference difference expected by chance Chi square as a statistical test

Chi-Square Test in SPSS Analyze  Descriptives  Crosstabs Chi-Square Test in SPSS

Chi-Square Test in SPSS Analyze  Descriptives  Crosstabs Click this to get the expected frequencies and residuals Click this to get bar chart of the results Chi-Square Test in SPSS

Chi-Square Test in SPSS

In lab: Gain experience using and interpreting Chi-square procedures Questions? Chi-squared test: https://www.youtube.com/watch?v=WXPBoFDqNVk (~12 mins) Chi-squared test: https://www.youtube.com/watch?v=SvKv375sacA (~38 mins) Chi-squared in SPSS: https://www.youtube.com/watch?v=wfIfEWMJY3s Chi-squared distribution: http://onlinestatbook.com/2/chi_square/distributionM.html Wrap up