Correlation and Regression

Transcription

1 Correlation and Regression Shepard and Feng (1972) presented participants with an unfolded cube and asked them to mentally refold the cube with the shaded square on the bottom to determine if the two arrows came together. Shepard and Feng told the participants to do this as quickly and as accurately as possible. The time to answer whether the arrows came together or not was measured in milliseconds (ms; 1 ms = 1/1000 of a second). Shepard and Feng also recorded the number of correct responses. Shepard and Feng found that the response time was a linear function of the sum of the number of squares that were carried along as each fold was mentally made. In the above example, a participant might start by making a mental fold on the left edge of the shaded square. One square (the leftmost square with an arrow on it) is carried along. Next, the participant might make a mental fold on the bottom edge of the shaded square. This time three squares are carried along (all of the squares below the shaded square.) Finally, the participant might make a mental fold at the top edge of the bottom square in the diagram. Only the bottom square is carried during that fold. Thus, the sum of the number of squares carried along for each fold is 5 (1 square in the first fold, 3 squares in the second fold, and 1 square in the last fold.) A similar study was performed with University of Dayton undergraduates. Their data is presented near the end of this handout. The data is also available as a SPSS data set at < Correlation 1. Step 1: Write the null and alternative hypotheses and specify the probability of making a Type I error: H 0 : ρ 0 H 1 : ρ > 0 This is a directional test because Shepard and Feng (1972) claim that response time should increase as the sum of the number of squares carried along for each fold increases. α =.05

2 2. Step 2: We will compare the reported p value to α. If p α, we will reject H 0 and conclude that the two variables are directly related to each other. 3. Step 3: Calculate r and its t: a. Open SPSS b. Get the data set at < c. Analyze Correlate Bivariate d. Move the two variables of interest (Squares and ) into the Variables: box e. Check the correct type of correlation coefficient. Since both of the variables are ratio scale, Pearson is the correct correlation coefficient f. Select whether the test is one- or two-tailed. Our hypothesis is directional, so this should be a one-tailed test

3 g. Click the Options button h. Check the Means and standard deviations option i. Click the Continue button j. Click the OK button k. The SPSS output viewer will open

4 l. The first part of the output is the descriptive statistics for each variable: This tells us the mean (4.63), standard deviation (2.50) and sample size (440) for the sum of the number of squares carried along for each fold. It also tells us the mean ( ), standard deviation ( ) and sample size (440) for the response times. m. The next part of the output is the correlation coefficient and its associated t test p value: Pearson s r for the sum of the number of squares carried along for each fold and the response time equals.681 (from the row with one of the variables and Pearson Correlation and the column with the other variable.) The associated t test has a p value equal to.000 (from the row with one of the variables and Sig. (1-tailed) and the column with the other variable.) 4. Step 4: Make a decision: If the p value (.000) is less than or equal to α (.05), reject H 0. In this example, we reject H 0 and conclude it is likely that there is a direct relation between the sum of the number of squares carried along for each fold and the response time. We would write: Pearson s r revealed a direct relation between the sum of the number of squares carried along for each fold and the response time, r =.681, n = 440, p =.000, one tailed. 5. Creating a scatterplot / scattergram: a. Graphs Chart Builder b. If this warning box appears, make sure that the level of measurement has been set for each variable in the Variable View:

5 The variables in this data set have had their level of measurement defined in SPSS. You can click the OK button to cancel the warning. c. The Chart Builder appears:

6 d. On the Gallery tab (it should be selected by default), click on the Scatter/Dot option in the Choose from: list:

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8 e. Drag the first type of scatterplot (I put a red rectangle around it above) into the large box above it f. Drag the variable that you are trying to predict ( in this example) to the Y-Axis? box

9 g. Drag the variable that you are trying to predict from (Squares in this example) to the X-Axis? box h. Click OK

10 i. The scatterplot appears in the output viewer: j. Double click the scatterplot to open the Chart Editor:

11 k. Click the Add Fit Line at Total button (I put a red rectangle around it above) l. Close the Chart Editor (click the X in the upper right corner, or File Close or Ctrl+F4) Regression 1. Analyze Regression Linear

12 2. Drag the variable you are predicting ( in this example) to the Dependent box 3. Drag the variable you are predicting from (Squares in this example) to the Independent(s) box

13 4. Click OK 5. The regression output appears in the SPSS output viewer. The last part of the output (Coefficients) is what we need for the slope and intercept of the regression line: 6. The slope is at the intersection of the column labeled B and the row labeled with the variable that you are predicting from (squares carried.) In this example, it equals The intercept is at the intersection of the column labeled B and the row labeled (Constant). In this example, it equals The regression equation is: ^ = X Sum of the of Squares Along for each Fold

14 Squares Squares Squares

15 Squares Squares Squares

16 Squares Squares Squares

17 Squares

18 Correlation and Regression 1. Sum the values of each variable. ΣX Squares = ΣX = 3,092, Sum the squared values of each variable. ΣX 2 Squares = 12,155. ΣX 2 = 31,141,831, Calculate the sum of squares for each variable. SS Squares = ΣX 2 Squares - (ΣX Squares ) 2 / n = 12, / 440 = SS = ΣX 2 (ΣX ) 2 / n = 31,141,831,107-3,092,337 2 / 440 = 9,408,767, For each row, multiply the value of Squares by the value of. Sum the product: ΣX Squares X = 17,762, Calculate the sum of the products: SP = ΣX Squares X (ΣX Squares X ΣX ) / n = 17,762,021 (2035 X 3,092,337) / 440 = 3,459, Calculate r: r = SP / ( SS Squares X SS ) = 3,459, / ( X 9,408,767,194) = Write the null and alternative hypotheses and specify α: H 0 : ρ 0 H 1 : ρ > 0 α = Calculate t: t = (r 2 / ((1 r 2 ) / df)) = ( / ( ) / 439) = t critical (439) (if the table does not contain the desired degrees of freedom, use the next smaller degree of freedom in the table. In this case, df = 120 is the next smallest degrees of freedom.) Because t observed = is greater than or equal to t critical 1.658, we will reject H Calculate the slope: SP / SS Squares = 3,459, / = Calculate the intercept: M slope X M Squares = 3,092,337 / X 2035 / 440 =