Srinivasulu Rajendran Centre for the Study of Regional Development (CSRD) Jawaharlal Nehru University (JNU) New Delhi India [email protected] Objective of the session To understand CORRELATION 1. What is the procedure to perform Correlation & Regression? 2. How do we interpret results? Identify the relationship between variables that we want to perform Scatter plot for outliers and type of relationship Monthly HH food Expenditure and HHSIZE Interpreting Correlation Coefficient r strong correlation: r > .70 or r < –.70 moderate correlation: r is between .30 & .70 or r is between –.30 and –.70 weak correlation: r is between 0 and .30 or r is between 0 and –.30 . GENERATE A SCATTERPLOT TO SEE THE RELATIONSHIPS Go to Graphs → Legacy dialogues→ Scatter/Dot → Simple Click on DEPENDENT “mfx”. and move it to the Y-Axis Click on the “hhsize”. and move it to the X-Axis Click OK Scatterplot might not look promising at first Double click on chart to open a CHART EDIT window use Options →Bin Element Simply CLOSE this box. Bins are applied automatically. BINS Dot size now shows number of cases with each pair of X, Y values DO NOT CLOSE CHART EDITOR YET! Add Fit Line (Regression) In Chart Editor: Elements →Fit Line at Total Close dialog box that opens Close Chart Editor window Edited Scatterplot Distribution of cases shown by dots (bins) Trend shown by fit line. Type of Correlation Bivariate Correlations. Partial Correlations Distances BIVARIATE CORRELATIONS In Bivariate Correlations, the relationship between two variables is measured. The degree of relationship (how closely they are related) could be either positive or negative. The maximum number could be either +1 (positive) or -1 (negative). This number is the correlation coefficient. A zero correlation indicates no relationship. Remember that you will want to perform a scatter plot before performing the correlation (to see if the assumptions have been met.) Objective We are interested in whether an monthly HH food expenditure was correlated with hhsize. Step 1 The Bivariate Correlations dialog box will appear List of Variables Right arrow button to add selected variable(s) Step 2 Select one of the variables that you want to correlate by clicking on it in the left hand pane of the Bivariate Correlations dialog box i.e mfx and hhsize Check the type of correlation coefficients that you require (Pearson for parametric, and Kendall’s tau-b and Spearman for non-parametric). Select the appropriate Test: Pearson’s correlation coefficient assumes that each pair of variables is bivariate normal and it is a measure of linear association. Two variables can be perfectly related, but if the relationship is not linear, Pearson’s correlation coefficient is not an appropriate statistic for measuring their association. Test of Significance: You can select two-tailed or one-tailed probabilities. If the direction of association is known in advance, select One-tailed. Otherwise, select Two-tailed. Flag significant correlations. Correlation coefficients significant at the 0.05 level are identified with a single asterisk, and those significant at the 0.01 level are identified with two asterisks. Click on the Options… button to select statistics, and select Means and SD and control the missing value by clicking “Exclude Cases pairwise. Click on the Continue button. Step 3 Click the OK button in the Bivariate Correlations dialog box to run the analysis. The output will be displayed in a separate SPSS Viewer window. SPSS Output of Correlation Matrix The Descriptive Statistics section gives the mean, standard deviation, and number of observations (N) for each of the variables that you specified. Descriptive Statistics Household size Monthly hh food expenditure (taka) Std. Mean Deviation 4.34 1.919 4411.25 2717.13 N 1237 1237 The correlations table displays Pearson correlation coefficients, significance values, and the number of cases with non-missing values (N). Correlations The values of the correlation coefficient range from -1 to 1. The sign of the correlation coefficient indicates the direction of the relationship Household (positive or negative). size The absolute value of the correlation coefficient indicates the strength, with larger absolute values indicating stronger relationships. Monthly hh food The correlation coefficients on the main expenditure diagonal are always 1, because each variable (taka) has a perfect positive linear relationship with itself. Pearson Correlation Sig. (1tailed) N Pearson Correlation Sig. (1tailed) N Monthl y hh food expend Househol iture d size (taka) ** 1 .608 .000 1237 ** .608 1237 1 .000 1237 1237 The significance of each correlation coefficient is also displayed in the correlation table. Correlations The significance level (or p- value) is the probability of obtaining results as extreme as the one observed. If the significance level is very small (less than 0.05) then the correlation is significant and the two variables are linearly related. If the significance level is relatively large (for example, 0.50) then the correlation is not significant and the two variables are not linearly related. Household size Monthly hh food expenditure (taka) Pearson Correlation Sig. (1tailed) N Pearson Correlation Sig. (1tailed) N Monthl y hh food expend Househol iture d size (taka) ** 1 .608 .000 1237 ** .608 1237 1 .000 1237 1237 Partial Correlations The Partial Correlations procedure computes partial correlation coefficients that describe the linear relationship between two variables while controlling for the effects of one or more additional variables. Correlations are measures of linear association. Two variables can be perfectly related, but if the relationship is not linear, a correlation coefficient is not a proper statistic to measure their association. Step 1 How to perform Partial Correl: SPSS Analyze –> Correlate –> Partial... You will be presented with the “Partial Correlations" dialogue box: Step 2 Click right click on variables and select “Display Variable” Click “Sort Alphabetically “ Step 3 Step 4 Select one of the variables that you want to correlate by clicking on it in the left hand pane of the Bivariate Correlations dialog box i.e mfx and hhsize In this case, we can see the correlation between monthly HH food expenditure and household size when head of education maintain constant. Test of Significance: You can select two- tailed or one-tailed probabilities. If the direction of association is known in advance, select One-tailed. Otherwise, select Two-tailed. Flag significant correlations. Correlation coefficients significant at the 0.05 level are identified with a single asterisk, and those significant at the 0.01 level are identified with two asterisks. Click OK to get results Step 5 As we can see, the positive correlation between mfx and hhsize when hh_edu is maintained constant is significant at 1% level (p > 0.00) Correlations Control Variables (sum) head_edu Household size Correlation Monthly hh food expenditure (taka) Monthl y hh food expend Househol iture d size (taka) 1.000 .606 Significance (1-tailed) . .000 df 0 1232 Correlation .606 1.000 Significance (1-tailed) .000 . df 1232 0 Hands-on Exercises Find out the correlation relationship between per capita total monthly expenditure and household size and identify the nature of relationship and define the reasons? Find out the correlation relationship between per capita total monthly expenditure and household size by controlling the village those who have adopted technology and not adopted tech? Find out the correlation relationship between per capita food expenditure and non-food expenditure by controlling district effect? [Hint: it is two tail why?] Distances This procedure calculates any of a wide variety of statistics measuring either similarities or dissimilarities (distances), either between pairs of variables or between pairs of cases. These similarity or distance measures can then be used with other procedures, such as factor analysis, cluster analysis, or multidimensional scaling, to help analyze complex data sets.