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1 Scatterplots Quantitative Research Methods: Introduction to correlation and regression Scatterplots can be considered as interval/ratio analogue of cross-tabs: arbitrarily many values mapped out in -dimensions Figure 1: Age 7, income, Brendan Halpin, Sociology, University of Limerick January 18 Income 7 units from left 3 units from bottom 8 1 Age 1 Age vs Income Summarising association simply Figure : Age and income, Wave 1 BHPS. Income We can see a lot of detail in a scatterplot, but sometimes we can summarise it in simple ways For instance the two variables may have a positive association: when one is high the other tends to be high, and vice AT UNIVERSITY versa OF LIMERICK AT Or a negative association: when one is high the other tends to be low 8 1 Age AT 3

2 Strong positive Figure 3: Fictional data displaying a strong positive linear relationship. Strong negative Figure : Fictional data displaying a strong negative linear relationship AT 1 AT AT Weak negative Figure : Fictional data displaying a weak negative linear relationship. 3 1 AT No relationship Figure : Fictional data displaying absence of a relationship AT - AT AT 7 8 AT AT 1 8

3 The correlation coefficient Deviations How does it work? Combines X deviations (X i X ) and Y deviations (Y i Ȳ ) i.e., compares each point with the mean for X and the mean for Y With positive association, cases below average on X tend to be below average on Y (and above average on X tend to be above average on Y) With negative association, cases below average on X tend to be above average on Y and vice versa With positive association below the mean, both X i X and Y i Ȳ are negative, so (X i X )(Y i Ȳ ) is positive With negative association, X i X and Y i Ȳ tend to have opposite signs, so (X i X )(Y i Ȳ ) is negative 9 Figure 7: X and Y deviations for correlation Y Y Y X X _ X X _ Y 1 Pearson Product-Moment Coefficient Pitfalls Pearson Product-Moment Coefficient (r) Range: 1 r +1 r = SXY SXX.SYY SXX = Σ(X X ) SYY = Σ(Y Ȳ ) SXY = Σ(X X )(Y Ȳ ) is not causality Absence of correlation is not absence of relationship: non-linearity r is a symmetric measure: r xy = r yx 11 1

4 Non-linear! Estimating correlations in Stata Figure 8: Fictional data displaying a strong non-linear relationship and a near-zero correlation t, t**+invnorm(rand(1)). pwcorr Income JobScore Hours, sig Income 1. Income JobScore Hours JobScore Hours Viewing the same correlations Regression analysis Regression Analysis: Fitting the best line through the scatter Very closely related to correlation, but treats one variable as dependent and the other(s) as explanatory, while correlation is asymmetric 1

5 Some geometry: equation of a line Predictive in intent Figure 9: The equation of a line: Y = a + bx 1. X = 1; Y = 1. 1 Y =.7 +.3*X. X = ; Y = Asymmetric: use X to predict Y PRE: implied causality (Proportional Reduction in Error: if we know X, we can guess Y better) Find the best a and b to summarise the data scatter: Best is defined as minimising the squared deviations between the observed data-points and the fitted line, hence often called least-squares regression Deviations are the vertical distance between the line and the observed data points. Very similar logic to the mean (minimise variance) Deviations again Predicted values Figure 1: Deviations from the line The line gives a predicted value of Y for each value of X : Ŷ = a + bx "Deviations" e is the residual or deviation. Y = Ŷ + e Y = a + bx + e That is, knowing X we predict or guess Y as a + bx In general this is more accurate that guessing Y as Ȳ, the mean: Proportionate Reduction in Error 19

6 Regression equation Pitfalls Regression equation: the estimate of Y, called Ŷ, depends on X : Ŷ = a + bx The regression slope b depends on SXY and SXX, the intercept a is calculated from b and the mean values of Y and X : b = SXY SXX a = Ȳ b X Like correlation, non-linear relationships may be missed Spurious relationships will fit just as well as real ones (e.g., if A affects B and A affects C, B and C will seem to be related and a regression line might fit well) Predicting outside the range of the data: the relationship we see only holds for the data we use, and it may well not hold for higher (or lower) values of X and Y SXX = (X i X ) SXY = (X i X )(Y i Ȳ ) 1 Fit Regression in Stata: How well does it fit? We use R to tell: ranges from : no relationship at all to 1: perfect relationship, all $Y$s are exactly equal to a + bx values from.7 up indicate quite a good relationship smaller values may indicate an interesting relationship In the case of bivariate regression (one independent variable), R is the same as r r (squared correlation coefficient).. reg Income JobScore Source SS df MS Number of obs = 7 F(1, ) = Model Prob > F =. Residual R-squared =.19 Adj R-squared =.199 Total Root MSE = 11.1 Income Coef. Std. Err. t P> t [9% Conf. Interval] JobScore _cons

7 Predicted regression line Multiple Regression Multiple explanatory variables Regression analysis can be extended to the case where there is more than one explanatory variable multivariate regression This allows us to estimate the net simultaneous effect of many variables, and thus to begin to disentangle more complex relationships Interpretation is relatively easy: each variable gets its own slope coefficient, standard error and significance The slope coefficient is the effect on the dependent variable of a 1 unit change in the explanatory variable, while taking account of the other variables Multiple Regression Example Multiple Regression Dichotomous variables Example: domestic work time may be affected by gender, and also by paid work time: competing explanations one or the other, or both could have effects We can fit bivariate regressions: or DWT = a + b PaidWork DWT = a + b Female We can also fit a single multivariate regression DWT = a + b PaidWork + c Female We deal with gender in a special way: this is a binary or dichotomous variable has two values We turn it into a yes/no or /1 variable e.g., female or not If we put this in as an explanatory variable a one-unit change in the explanatory variable is the difference between being male and female Thus the c coefficient we get in the DWT = a + b PaidWork + c Female regression is the net change in predicted domestic work time for females, once you take account of paid work time. The b coefficient is then the net effect of a unit change in paid work time, once you take gender into account. 7 8

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