Association Between Variables Measured at the Interval-Ratio Level: Bivariate Correlation and Regression

Transcription

1 Association Between Variables Measured at the Interval-Ratio Level: Bivariate Correlation and Regression Last couple of classes: Measures of Association: Phi, Cramer s V and Lambda (nominal level of measurement) Gamma (ordinal level of measurement) Today: what if: interval/ratio level of measurement Assignment 5 is now posted (due next week, last day of classes) Next Week: Review for final exam EXAM IS ON: Saturday: Dec 10th, 7:00 p.m LH 100

2 Introduction: Interval/ratio level of measurement Scores are actual numbers and have a true zero point and equal intervals between scores E.g. Age (in years); Income (in dollars); Education (in years) Weight (in pounds) Hours worked (hours) etc.

3 Introduction: Interval/ratio level of measurement Example: Two variables, Age (0-100 years) & Family Size (1-10) Sometimes bivariate tables are impractical with interval/ratio variables (10 rows X 100 columns) Rather than working with Bivariate Tables, we work with scattergrams columns

4 Example of a Hypothetical Scattergram Showing the Relationship Between X and Y Across 150 countries (cases) Case 1 Case Case 3 Case 4 (9,105) Case 150 (50, 30) Retaining as much information as possible Regression uses directly all of this detailed information!!

5 Regression is all about representing a relationship linearly.. Regression fits a straight line that best represents the data Y = a + bx Where: a is the y intercept b is the slope

6 Assume our regression line (Y= a + bx) is: Y = X Slope (b = 1.4) Y intercept (a = 3) when X=0, Y=3 For every 1 unit increase in the illiteracy rate, we can expect a 1.4 unit increase in the dependent

7 Positive and negative associations are possible Positive associations are represented by positive slopes In this case, the higher a society scores in terms of the illiteracy rate, the higher we would predict the infant mortality rate

8 Negative associations are possible -> negative slope In this case, the higher the percentage with access to drinking water the lower the observed infant mortality rate

9 An absence of an association has a slope of zero Alcohol consumption (Y) Height (X)

10 In addition to the direction (positive or negative), we are also interested in both the strength and significance of relationships.. Linear relationships vary in terms of the strength of the associations involved: Example: the right graph portrays a much stronger association The greater the cases are clustered around the regression line, the stronger the relationship. Based on the regression slope, we can calculate an additional statistic: Pearson s R (also called the correlation coefficient) which serves as a measure of association for interval variables (details forthcoming) Like Gamma, ranges form -1.0 thru +1.0

11 Regression is all about representing a relationship linearly..

12 Begin with the slope: How do we obtain the Regression Line: y = a + bx? We WORK WITH THE FOLLOWING FORMULA in SOLVING FOR b Computational (working) formula for the slope (Formula 13.3) b nxy nx ( X )( Y ) ( X )

13 Secondly, calculate (a) The Y intercept (a) is computed using Formula 13.4: If we have our slope b & the mean score of X & the mean score of Y, it is easy to obtain the Y intercept (a) With this, you now have our regression line.. Y = a + bx

14 How do we obtain Pearson s r? ] ) ( ][ ) ( [ ) )( ( Y Y n X X n Y X XY n r If you have calculated b, you can also easily calculate this measure of association using:

15 Pearson s r Like Gamma, r is our measure of association and it varies from to Pearson s r is a measure of association for Interval- Ratio variables. In interpreting the strength of r, use the same table as the one we had for gamma. As will be demonstrated, we can also test r for significance, using the familiar 5 step model.

16 Practical Example The computation and interpretation of a, b and Pearson s r will be illustrated using the following example: Problem: The variables are: Voter turnout (Y) is the dependent variable. Average years of school (X) is the independent variable. The sample is 5 cities. This is only to simplify the calculation. A sample of 5 is actually very small NOTE: computer programs do this with 1000 s of cases

17 Data for Problem: City X Y A B The scores on each variable are displayed in table format: Y = % Turnout X = Years of Education C D E NOTE: VERY IMPORTANT TO SET YOUR DEPENDENT (Y) AND INDEPENDENT VARIABLES (X) UP CORRECTLY!! nxy ( X )( Y ) b nx ( X ) a y bx r [ nx nxy ( X ( X ) ][ ny )( Y ) ( Y ) ]

18 1. Make a Computational Table: X Y X Y XY X = 6.5 Y = 318 X =78.15 Y = 0374 XY =

19 1. Make a Computational Table: X Y X Y XY X = 6.5 Y = 318 X =78.15 Y = 0374 XY =

20 1. Make a Computational Table: X Y X Y XY X = 6.5 Y = 318 X =78.15 Y = 0374 XY =

21 1. Make a Computational Table: X Y X Y XY X = 6.5 Y = 318 X =78.15 Y = 0374 XY = Use above to calculate the mean of X and Y: Y Y / n 318/ X X / n 6.5/ 5 1.5

22 3. Next calculate slope: b nxy nx ( X ( X )( Y ) )

23 X Y X Y XY X = 6.5 Y = 318 X =78.15 Y = 0374 XY = nxy ( X )( Y ) 5(3986.4) (6.5)(318) b nx ( X ) 5(78.15) (6.5)

24 4. Next, calculate a (y intercept) a Y bx We had originally documented that: X 1.5 Y 63.6 And we had just documented that b = 1.67, so: a Y bx (1.5)

25 Our regression line: Y Y a bx ( X ) Interpretation: The y intercept (a) is the point at which the regression line crosses the Y-axis (when X is equal to 0, Y is equal to ) Slope (b) indicates that: for every unit increase in X, Y increases by This means that for 1 additional year of schooling, voter turnout goes up by 1.67%.

26 5. Calculate the correlation coefficient r r [ nx nxy ( X ( X ) ][ ny )( Y ) ( Y ) ]

27 X Y X Y XY X = 6.5 Y = 318 X =78.15 Y = 0374 XY = r [ nx nxy ( X )( Y ) ( X ) ][ ny ( Y ) 5(3986.4) (6.5)(318) [5(78.15) (6.5) ][5(0374) (318) ].984

28 Interpret Pearson s r An r of 0.98 indicates an very strong relationship between years of education and voter turnout for these five cities (use the table given in with gamma to estimate strength)

29 6. Testing r for significance: We can test the relationship between % turnout and years of education (represented by Pearson s r) for significance using the 5 step model and the following formula: t r Degrees of Freedom = N- obtained 1 r & n is sample size In this case we work with the t distribution, as n is small: Note: the t distribution is one and the same as the z distribution when n>100 Again we are working with a sampling distribution (in this case, of our Pearson s r ) n

30 Step 1: Assumptions Random sample & additional assumptions The relationship between X and Y is linear. Interval/ratio measurement

31 Step : Null and Alternate Hypotheses: H o : ρ = 0.0 H 1 : ρ 0.0 (Note that ρ (rho) is the population parameter, while r is the sample statistic.) Step 3: Sampling Distribution and Critical Region: Sampling Distribution = t-distribution Alpha =.05 DF = n - = 5 - = 3 t critical = 3.18 (you can find this in the T table (appendix) Note: If n is greater than 100, we are essentially using the Z distribution

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33 Step 4. Computing the Test Statistic: Use Formula 13.8 in Healey t obtained r n r 1 (.984) 9.53 Step 5. Decision and Interpretation: T obtained = 9.53 > t critical = 3.18 Reject H o. The relationship between % turnout and years of schooling is significant.

34 Always include a brief summary of your results: There is a very strong, positive relationship between % voter turnout and years of schooling for the five cities. As years of schooling increase, the % of voter turnout goes up. The relationship is significant (t=9.53, df=3, α =.05).

35 ONE ADDITIONAL POINT: We can use the regression equation for prediction. Find the Regression Line: Y a bx ( X ) Note: you can now use it for prediction purposes.. For prediction: Suppose years of schooling = 10 years Then, Y = (10) = We would predict that when average years of education is 10 years, the voter turnout would be 31.97%

36 LET US GO THRU ONE MORE EXAMPLE OF REGRESSION.. Looking at the relationship between: Number of children in the household Number of Husband s hours worked Which is the likely dependent?? What s the nature of the relationship between the two variables? Direction and strength?? Significance? Get X Y XY

37 1. Make computation table, and. calculate the means of Y and X 15-37

38 3. Calculate the slope Regression Analysis (continued) 15-38

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40 Regression Analysis (continued) 4. Calculate the Y intercept 15-40

41 Interpret For the relationship between number of children and husband s housework: b (slope) =.69 a (Y intercept)= 1.49 A slope of.69 means that the amount of time a husband contributes to housekeeping chores increases by.69 (less than one hour per week) for every unit increase of 1 in number of children (for each additional child in the family). The Y intercept means that the regression line crosses the Y axis at Y = 1.49 (or the value of Y when X is 0)

42 15-4 Pearson s r is a measure of association for two interval-ratio variables. 5. Calculate Pearson s r ] ) ( ][ ) ( [ ) )( ( Y Y n X X n Y X XY n r

43 Pearson s r: An Example The quantities displayed earlier can again be directly substituted directly into formula to calculate r for our sample problem 15-43

44 Pearson s r: An Example The quantities displayed earlier can again be directly substituted directly into formula to calculate r for our sample problem

45 Interpreting Pearson s r Use the guidelines stated in Table (for gamma) as a guide to interpret the strength of Pearson s r. As before, the relationship between the values and the descriptive terms is arbitrary, so the scale in Table is intended as a general guideline only: 15-45

46 Interpreting Pearson s r (continued) An r of 0.50 indicates a strong positive linear relationship between the variables. As the number of children in a family increases, the hourly contribution of husbands to housekeeping duties also increases. BUT: IS IT SIGNIFICANT???? 15-46

47 6. Testing Statistical Significance of r Step 1: Assumptions Random sample & additional assumptions The relationship between X and Y is linear. also: interval/ratio level of measurement

48 Step : Null and Alternate Hypotheses: H o : ρ = 0.0 H 1 : ρ 0.0 (Note that ρ (rho) is the population parameter, while r is the sample statistic.) Step 3: Sampling Distribution and Critical Region: Sampling Distribution = t-distribution Alpha =.05 DF = n - = 1 - = 10 t critical =.8 (from the t distribution table)

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50 Step 4. Computing the Test Statistic: Use Formula for test statistic: t obtained r n r 1 (.5) Step 5. Decision and Interpretation: T obtained = 1.86 < t critical =.8 Can t Reject H o. The relationship between the two variables is not significant!!

51 Always include a brief summary of your results: While there initially appeared to be a strong positive relationship between the two variables, we found that the relationship was non-significant (t=1.86, df=10, α =.05).