Quantitative Methods Topic 9 Bivariate Relationships

1. Quantitative MethodsTopic 9Bivariate Relationships

2. Outline • Crosstab (Exploring the relationship between two categorical variable). • Correlation (Exploring the relationship between two continuous variables, typically)

3. Data file • YR12SURV2.SAV YR12SURVEYCODING2.DOC (Questionnaire) Holland2fory12data.doc

4. RANKMS and RANKINV • RANKMS is a constructed variable ranking informants on the amount of time spent on Maths and Science • A high rank (e.g. 3) means the informant spent a lot of time on Maths and Science • A low rank (e.g. 1) means the informant spent very little or no time on Maths and Science. • RANKINV is a similar rank type variable focused on investigative interests: e.g. informant interested in laboratory work. • A high rank (e.g. 4) means the informant was very interested • A low rank (.e.g. 1) means the informant was least interested. • These are ordinal variables.

5. Relationships between two categorical variables Example of research questions • Is there a relationship between gender and student investigative interest? • Are males more likely to be interested in investigative activities than females? • Is the proportion of males in each of the investigative level the same as the proportion of females?

6. Variables • To answer the research question in the example above we will have to do crosstabulations between two variables • Gender • RANKINV

7. Hypothesis of independence • There is no association between the two variables gender and RANKINV • There is no difference in the proportion of females and males in each of the categories (levels) of investigative interest

8. How to do crosstabulations in SPSS • From the DATA menu select ANALYSE then Descriptive Statistics then Crosstabs • Move GENDER into the Column(s) window and RANKINV into the Row(s) window • Open the Statistics window and tick Chi-square Continue to close • Open the Cells window, under Counts tick observed, under Percentages, tick Column, then click Continue • Click OK in the Crosstabs window to run

9. Screen

10. Output • There are two tables in the output that are important for us: • Table 1: Crosstab • Table 2: Chi-square

11. Table 1: Crosstab of RankInv by Gender

12. Interpreting Association in the Table • We can compare the column percentagesalong the rows and calculate the percentage point difference to see (in this case) whether females differ from males at each ‘level’ of interest • In the rankinvestgtv by Gender crosstabulation, for example, 30.7% of females were in category 1 (Very low investigative interest) compared with 19.6% of males, giving a percentage point difference of 11.1. • Similarly, there is a difference of 10.7 percentage points in the number of males having very high investigative interest compared with females

13. Table 2: Chi-square Statistics generated by Crosstabs Pearson Chi-square value, degree of freedom and significant level This helps us to check if expected counts less than 5

14. Tests of Statistical Significance for Tables • Chi-square used to test the null hypothesis that there is no discrepancy between the observed and expected frequencies or there is no association between row and column variables • Chi-square based statistics can be used independently of level of measurement. • If chi-square is significant (say Asymp. Sig. <0.05) then we reject the null hypothesis and conclude that the data show some associationcompared with a (hypothetical) table in which the observed frequencies were determined solely by the separate distributions of the two crosstabulated variables (the ‘marginal distributions’) • If chi-square is not significant (say Asymp. Sig. >0.05) then we accept the null hypothesis and conclude that the data show no associationcompared with a (hypothetical) table in which the observed frequencies were determined solely by the separate distributions of the two crosstabulated variables (the ‘marginal distributions’)

15. Assumptions • Random samples • Independent observations: the different groups are independent of each other • The lowest expected frequency in any cell should be 5 or more

16. Chi-square Statistics-limitations • Chi-Square measures are sensitive to sample size

17. Interpretation of output from chi-square • The note under the table shows that you have not violated one of the assumptions of chi-square concerning ‘minimum expected cell frequency’ • Pearson chi-square value: • at 18.5 for 3 degrees of freedom Chi-square is highly significant • probability of this level of association occurring by chance is less than 0.001. • Degree of freedom=(r-1)(c-1) where r and c are number of categories in each of the two variables. • Conclusion: males are more likely than females to be interested in investigative activities.

18. Class activity 1: Produce a similar table using GENDER by RANKMS

19. Summary of analyses of association • RANKINV and RANKMS are 4 and 3 categories (respectively) ordinal variables constructed, respectively, from the total score on Investigative interests and the proportion of curriculum time spent in Maths/Science • Gender heads the columns, interest and curriculum participation in Maths/Science form the rows • (thus, by convention, gender is the explanatory or independent variable, interest or curriculum participation the responseordependentvariables)

20. Correlations • Strengths of relationships between two variables.

21. Correlation Examples of Research questions • Is there a relationship between student achievement in mathematics and English language? • Is there a relationship between parents’ incomes and children VCE results ? • Is there a correlation between SES and achievement ? • How strong are these relationships?

22. Assumptions (1) • Scores are obtained using a random sample from populations • Independence of observations • The distribution of the variable(s) involved is normal • Homoscedasticity: the variance of the dependent variable is the same for values of X (residual variance, or conditional variance) • Linearity: The relationship between the two variables should be linear. • Related pairs: both pieces of information must be from the same subjects

23. Data Set • Vietnam Data Set vnsample2.sav

24. Scatter plot

25. Producing a Scatterplot • GRAPHS-SCATTER-SIMPLE-DEFINE • Select MEASUREMENT score (pmes500) to make this the Y variable • Select NUMBER score (pnum500) to make this the X variable. • Click OK • The scatterplot should appear in the OUTPUT window.

26. Scatter plot

27. Interpretation of Scatter plot • Step 1: Checking for outliers • Step 2: INSPECTING THE DISTRIBUTION OF DATA POINT: • Are the data points scattered all over. • Are the data points neatly arranged in a narrow cigar shape • Could we draw a straight line through the main cluster of points or would a curved line better represents the points • Is the shape of the cluster even from one end to other. (if it starts off narrow and then gets wider, The data may violate the assumption of homoscedasticity: at different value of X, variability of Y is different) • Step 3: Determining the direction of the relationship between the two variables: positive or negative correlations

28. Direct Relationship • When values on two variables tend to go in the same direction, we call this a direct relationship. • The correlation between children’s ages and heights is a direct relationship. • That is, older children tend to be taller than younger children. • This is a direct relationship because children with higher ages tend to have higher heights.

29. Inverse Relationship • When values on two variables tend to go in opposite directions, we call this an inverse relationship. • The correlation between students’ number of absences and level of achievement is an inverse relationship. • That is, students who are absent more often tend to have lower achievement. • This is an inverse relationship because children with higher numbers of absences tend to have lower achievement scores.

30. Scatter plot

31. How to run correlation • Highlight ANALYSE, CORRELATE, BIVARIATE • Copy THE TWO VARIABLES INTO VARIABLES box • Check that PEARSON box (two continuous variables- see the notes for other variable types) and the 2 tail box • Click OK

32. OUTPUT AND INTERPRETATION This is correlation coefficient (r) Number of cases Step 1: Checking information about sample size Step 2: Determining the directions and strengths of the relationships Step 3:Calculating the coefficient of determination (r2) Step 4: Assessing the significance This is the p value

33. Correlation Coefficient • The relationship between two variables may be expressed with a number between -.1.00 and 1.00. This number is called a correlation coefficient. • The closer the correlation coefficient is to 0.00, the lower the relationship between the two variables. The closer the coefficient is to 1.00 or -1.00 the higher the relationship. • According to Cohen (1988) • R=.10 to .29 or R=-.10 to -.29: Small • R=.30 to 0.49 or R=-.30 to -0.49 Medium • R=.50 to 1.00 or R=-.50 to -1 Large

34. Some caveats about correlation and scatter plots - 1 • Make a scatter plot of Measurement score against Number score again. • This time, double click on the plot to get into Chart Editor. • Change both X and Y axis scales to have a minimum of 200 and a maximum of 750. • Does the strength of the relationship look weaker in this graph as compared to the one where the min is 0 and max is 1000?

35. Some caveats about correlation and scatter plots - 2 • Be aware that judging the strength of relationship based on visual perception of scatter plot could be flawed, as the scale of the plots can make a difference.

36. Some caveats about correlation and scatter plots - 3 • Create a new variable pmes10 using Transform  compute new variable such that pmes10=pmes500/100 +5 • That is, we have transformed the measurement score to have a mean of 10 and a standard deviation of 1. • Compute the correlation between pmes10 and pnum500. • How does this correlation compare with the correlation between pmes500 and pnum500?

37. Some caveats about correlation and scatter plots - 4 • Compute the correlation between pmes500 and pnum500, but only for scores between 300 and 600. • You can do this by selecting a sample • Data  Select cases  If condition is satisfied: pnum500 > 300 and pnum500<600 and pmes500>300 and pmes500<600 • How does the correlation compare with that from the full sample?

38. Some caveats about correlation and scatter plots - 5 • This link shows TIMSS and PISA 2003 maths country mean scores for 22 countries. TIMSSandPISA 2003.doc • Plot a scatter graph between TIMSS and PISA scores • Compute the correlation • Repeat without Tunisia and Indonesia.

39. Coefficient of determination • is the squared correlation coefficient, and • represents the proportion of common variation in the two variables

40. Proportion of variance explained • The proportion of variance explained is equal to the square of the correlation coefficient • If the correlation between alternate forms of a standardized test is 0.80, then (0.80)2 or 0.64 of the variance in scores on one form of the test is explained or associated with variance of scores on the other form • That is, 64% of the variance one sees in scores on one form is associated with the variance of scores on the other form. Consequently, only 36% (100% – 64%) of the variance of scores on one form is unassociated with variance of scores on the other form.

41. Presenting the results for Correlation • Purpose of the test • Variables involved • r values, number of cases, p value • R-square • Interpretation

42. Class activity 1 • What are the correlations between three dimensions of mathematics: number, measurement and space? • Is it true that students who perform well in number strand also doing well in measurement and space strands? • Datafile: VNsample2.sav • Variables: student scores in measurement, number and space

43. Regression line

44. Producing a Scatterplot with regression line • GRAPHS-SCATTER-SIMPLE-DEFINE • SelectMEASUREMENT 500 SCORE IN PUPIL MATH to make this the Y variable • Select NUMBER 500 SCORE IN PUPIL MATH and to make this the X variable. • Click OK • The scatterplot should appear in the OUTPUT window. • Double click anywhere in the scatter plot to open SPSS CHART EDITOR, Click on ELEMENTS< FIT LINE AT TOTAL, make sure that LINEAR is selected.

45. Regression line

46. Regression line

47. Linear Regression Example of research questions • To what extent can student achievement in Vietnamese language predict student achievement in mathematics? • How well family income (or wealth) can predict student performance? • How well university entrance scores can predict student success in University?

48. Regression equation y= a+bx+e • y and x are the dependent and independent variables respectively • a is the intercept (the point at which the line cuts the vertical axis. • b is the slope of the line or the regression coefficient. • e is error term.

49. How to RUN REGRESSION • Highlight ANALYSE, REGRESSION, LINEAR • Copy THE CONTINOUS DEPENDENT VARIABLE INTO DEPENDENT box • Copy THE INDEPENDENT VARIABLE INTO INDEPENDENT box • For METHOD, make sure that ENTER is selected. • Click STATISTICS and tick on ESTIMATES, MODEL FIT AND DESCRIPTIVES, then CONTINUE • Click on OPTIONS, then INCLUDE CONSTANT IN EQUATION, EXCLUDE CASES PAIRWISE, then CONTINUE • Click OK

50. An example for regression • Research question: To what extent that student scores in reading can predict student scores in Mathematics? • Datafile: VNsample2.sav • Variables: Reading 500 scores, Mathematics 500 scores achievement