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Quantitative Data Analysis II: Correlation and Simple Linear Regression SI0030 Social Research Methods Week 6 Luke Sloan SI0030 Social Research Methods Week 6 Luke Sloan

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Introduction Last Week – Recap Correlation How To Draw A Line Simple Linear Regression Summary

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Last Week - Recap Hypotheses Probability & Significance (p=<0.05) Chi-square test for two categorical variables t-test for one categorical and one interval variables What about a test for two interval variables?...

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Correlation I Calculates the strength and direction of a linear relationship between two interval variables e.g. is there a relationship between age and income? Measured using the Pearson correlation coefficient (r) Data must be normally distributed (check with a histogram) If not normally distributed use Spearman’s Rank Order Correlation (rho) - consult Pallant (2005:297)

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Correlation II ‘r’ can take any value from +1 to -1 +/- indicates whether the relationship is positive or negative +1 or -1 is a perfect linear relationship, but usually it is not this clear cut Rule of thumb: – +/- 0.7 = a strong linear relationship – +/- 0.5 = a good linear relationship – +/- 0.3 = a linear relationship – Below +/- 0.3 = weak linear relationship – 0 = no linear relationship Alternatively: - +/- 0.10 to 0.29 = weak - +/- 0.30 to 0.49 = medium - +/- 0.50 to 1.00 - strong Alternatively: - +/- 0.10 to 0.29 = weak - +/- 0.30 to 0.49 = medium - +/- 0.50 to 1.00 - strong

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Correlation III Positive relationship Negative relationship No relationship Positive Relationship Positive Relationship No Relationship No Relationship Negative Relationship Negative Relationship Formulate hypotheses and use scatter plots!

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Correlation IV H 1 = There is a relationship between Age and the number of years a candidate has been a member of a political party H 0 = There is no relationship between Age and the number of years a candidate has been a member of a political party What do you think?

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Correlation V Is this normal? Just to prove a point…

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Correlation VI Correlations What was your age last birthday Number of years a party member What was your age last birthday Pearson Correlation1.425 ** Sig. (2-tailed).000 N44811874 Number of years a party member Pearson Correlation.425 ** 1 Sig. (2-tailed).000 N18741936 **. Correlation is significant at the 0.01 level (2-tailed). Perfect correlation against itself (obviously!) and number of cases in analysis Pearson’s Correlation Coefficient is r=0.43 – medium/good positive linear relationship Significance for correlation is problematic (highly dependent on sample size) – report p-value but ignore level of significance

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Correlation VII Don’t forget to refute or accept the null hypothesis and discuss the relationship Correlation is not causation! The relationship between the number of years a candidate has been a member of a party and candidate age was explored using Pearson’s correlation coefficient. Both variables were confirmed to have normal distributions [?] and a scatter plot revealed a linear relationship. There was a medium-strength, positive relationship between the two variables (r=0.43, n=4481, p<0.05)... [go on to explain the relationship in detail]

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How To Draw A Line I Correlation is indicative of a relationship, but it does not allow us to quantify it What if we wanted to explain how an increase in age leads to an increase in years of party membership? What if we wanted to predict years of party membership based only on age? Correlation is indicative of a relationship, but it does not allow us to quantify it What if we wanted to explain how an increase in age leads to an increase in years of party membership? What if we wanted to predict years of party membership based only on age? The line of best fit is a predictive – it is the regression line!

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How To Draw A Line II The regression line allows us to predict any given value of y when we know x i.e. if we know the age of a candidate we can predict how long they are likely to have been a member of a political party Another (more useful!) example would be years in education and income Using a regression line we can predict someone’s income based on the number of years they have been in education Assumes a causal relationship – that income is ‘caused’ by years in education

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How To Draw A Line III But… we don’t simply look very closely at the line and the axis of the scatter plot because the regression line can be written as an equation: y = a + b x ‘y’ represents the dependent variable (what we are trying to predict) e.g. income ‘a’ represents the intercept(where the regression line crosses the vertical ‘y’ axis) aka the constant ‘b’ represents the slope of the line (the association between ‘y’ & ‘x’) e.g. how income changes in relation to education ‘x’ represents the independent variable (what we are using to predict ‘y’) e.g. years in education

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y = 0 + 1x y = 0 + 2x y = 0 + 0.5x x axis y axis y = 0 + 0.25x y = 1 + 1x What about… How To Draw A Line IV

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If we know the slope (b) and the intercept (a), for any given value of ‘x’ we can predict ‘y’ Preconditions: Simple Linear Regression EXAMPLE: predicting income (y) in thousands (£) from years in education (x) Intercept (a) = 4 Slope (b) = 1.5 For someone with 10 years of education Equations: y = a + bx Income = intercept + (slope*years in education) Income = 4 + (1.5*10) = 19 (£19,000) Or…

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Simple Linear Regression II Assumptions – Interval level data – Linearity between ‘x’ and ‘y’ – Outliers (check scatter plot) – Sample size = 100+? R 2 measure of ‘model fit’ – Literally the Pearson’s correlation coefficient squared – R 2 tells us how much of the variance in the dependent variable is explained by the independent variable e.g. how much of the variance in income can be explained by age – Expressed as a percentage (1.0 = 100%, 0.5 = 50% etc)

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Simple Linear Regression III H 0 = There is no relationship between Age and the number of years a candidate has been a member of a political party H 1 = There is a relationship between Age and the number of years a candidate has been a member of a political party H 0 = There is no relationship between Age and the number of years a candidate has been a member of a political party H 1 = There is a relationship between Age and the number of years a candidate has been a member of a political party H 2 = As the age of a candidate increases, so will the number of years that they have been a party member ‘Years as Party Member’ = intercept + (slope * ’Age’)

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Simple Linear Regression IV Model Summary Model RR SquareAdjusted R Square Std. Error of the Estimate 1.425 a.181.18011.995 a. Predictors: (Constant), What was your age last birthday ANOVA b Model Sum of SquaresdfMean SquareFSig. 1Regression59446.0851 413.170.000 a Residual269339.6961872143.878 Total328785.7811873 a. Predictors: (Constant), What was your age last birthday b. Dependent Variable: Number of years a party member Pearson’s correlation coefficient (same value!) 18% of variance in party membership (y) explained by age (x) This tests the hypothesis that the model is a better predictor of party membership than if we simply used the mean value of party membership p<0.05 so the regression model is a significantly better predictor than the mean value

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Simple Linear Regression V Coefficients a Model Unstandardized Coefficients Standardiz ed Coefficient s tSig. BStd. ErrorBeta 1(Constant)-6.8991.156 -5.966.000 What was your age last birthday.418.021.42520.327.000 a. Dependent Variable: Number of years a party member This is the intercept (a) y = a + b x This is the slope (b) p<0.05 so ‘Age’ has a significant effect on ‘Party Membership’ ‘Party Membership’ = -6.9 + (0.42 * ’Age’) A one unit increase in age will result in an increase in party membership of 0.42 Or…

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Simple Linear Regression VI … and this is what we saw in the original scatter plot! The ‘regression line’ will intercept the verticle (y) axis at -6.9 The ‘regression line’ rises by 0.42 on the verticle axis (y) for every one unit increase on the horizontal axis (x) The R 2 value is low because of the fanning effect (remember the histograms!)

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Summary How to describe and quantify the relationship between two interval variables Correlation – the strength and direction of the association Regression – the causal and quantified effect of an independent on a dependent variable

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