ࡱ> dfc~WE@ bjbj .NNN8N, P2Q&BV"dVdVV $R ^] ! dVVq4@ 0dVLV ÖVY@VQ o'0N= V0xhvhh1 2D:d(D:Correlation & regression We often want to know whether various variables are linked, i.e., correlated. This can be interesting in itself, but is also important if we want to predict one variables value given a value of the other. Important to remember correlation is not an indication of causality; to infer causality we need to manipulate an independent variable and observe the effect this has on a dependent variable. Observation clouds:  SHAPE \* MERGEFORMAT  SHAPE \* MERGEFORMAT  Covariance: Do two variables change together?  EMBED Equation.3  Recall that variance is:  EMBED Equation.3  So its pretty similar, except were multiplying two variables, not just squaring one.  EMBED Equation.3  So, If when X increases so does Y cov(x,y) will be positive If when X increases Y decreases cov(x,y) will be negative If there is no constant relationship cov(x,y) will be zero For example:  EMBED MSGraph.Chart.8 \s   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 ( EMBED Equation.3 )( EMBED Equation.3 )03-30022-1-1134010401-3-366339 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  But what does this number tell us? Nothing, really, as  EMBED Equation.3  So we can only compare covariances between different variables to see which is greater. Or, we could standardize this measure, thus obtaining a more intuitive measure of correlation magnitude. Correlation: Pearson s R Standardize by adding the standard deviations to the equation:  EMBED Equation.3  !  EMBED Equation.3    EMBED Equation.3   EMBED Equation.3  Important: each xi goes with a specific yi Why? Example: changing just two points  EMBED MSGraph.Chart.8 \s   EMBED MSGraph.Chart.8 \s   EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 * EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 * EMBED Equation.3 03-1.50003-1.50022-0.5-0.50.2520-0.5-1.50.753400.503400.50400.5-1.5-0.75420.5-0.5-0.25661.51.52.25661.51.52.25 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  A limitation of r: it is very sensitive to extreme values. Example: xy 1122334450 Note: r is actually  EMBED Equation.3 . OK, so when r = 1 or r = (-1) we have a perfect linear relationship: y = ax + b But here in the real world, this never happens. Theres always a cloud of observations. So to understand the relationship between two variables, we want to draw the best line through the cloud find the best fit. This is done using the principle of least squares Regression  y = ax + b This is true for a sample. Like in all statistical methods, we want to make inferences about the population. So,   EMBED Equation.3   SHAPE \* MERGEFORMAT   EMBED Equation.3  (our model) Obviously, the stronger the correlation between x and y, the better the prediction; this is expressed in both parameters:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  A different, simpler way to write this:  EMBED Equation.3  Its easy to see why if theres no correlation, we will simply predict the average of y for any x. The larger the correlation, the greater the regression lines slope. In any case, the average of the predicted values will always equal the average of the true values:  EMBED Equation.3  (so  EMBED Equation.3  is an unbiased estimator of  EMBED Equation.3  ). The variance of the predicted values:  EMBED Equation.3  So this variance is always smaller than the true variance (as the true variance is multiplied by a fraction). Furthermore:  EMBED Equation.3  r-squared is the explained variance! It tells us what fraction of the general variance can be attributed to the model. Therefore: True variance = predicted variance + error variance  EMBED Equation.3  or:  EMBED Equation.3   Is the model significant? (do we get a significantly better prediction using it than we do by just predicting the mean?) This is where we see why it is similar to ANOVA*: SSTotal = SSRegression + SSError  EMBED Equation.3  * In a one-way ANOVA, we have SSTotal = SSBetween + SSWithin  EMBED Equation.3  From the SS we can derive MS dividing each SS by its degrees of freedom: MSRegression = SSRegression / 1 MSError = SSError / N-2 Statistical significance test:  EMBED Equation.3  Alternatively (as F is the square of t):  EMBED Equation.3  Important assumptions: Normal distributions Constant variances Independent sampling no autocorrelations  ~ N(0,2) No errors in the values of the independent variable All causation in the model is one-way (not necessary mathematically, but essential for prediction) The regression model:  EMBED Equation.3  The regression model in GLM terms:  EMBED Equation.3  So:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  And in matrix notation:  EMBED Equation.3   EMBED Equation.3  PAGE  PAGE 17 X Y Positive correlation Y X Y X Negative correlation No correlation  EMBED 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` a b z $a$gd z 0dh]^0gd$x z dh]gdm8         , - . / 0 1 D E ˻׬{lW{{HjlB h2CJUVaJ)j`h2h2CJ0EHUaJ0mH sH jdB h2CJUVaJjhCJ0UaJ0mH sH hCJ0aJ0mH sH )jh2h2CJ0EHUaJ0mH sH jTB h2CJUVaJjh%|CJ0UaJ0mH sH hm8CJ0aJ0mH sH h%|CJ0aJ0mH sH h2CJ0aJ0mH sH jh2CJ0UaJ0mH sH E F G H I \ ] ^ _ ` b y z { ο΢wfZKjB huJCJUVaJjhCJ0UaJ0!jhhCJ0EHUaJ0jB hCJUVaJjhCJ0UaJ0h{#CJ0aJ0hCJ0aJ0hCJ0aJ0)jeh2h2CJ0EHUaJ0mH sH jsB h2CJUVaJhCJ0aJ0mH sH jhCJ0UaJ0mH sH )jh2h2CJ0EHUaJ0mH sH  ˿vrcrSrhbhE 5CJ4aJ4mH sH hbhE CJaJmH sH hE hA hE CJaJmH sH h{q0JmHnHuh{qh# h#0Jjh#0JUhCJ0aJ0mH sH hhB*phhyCJ0aJ0mH sH h%|h{#CJ0aJ0mH sH h{#CJ0aJ0jhCJ0UaJ0!jLhhuJCJ0EHUaJ0 !!!!/!0!h]hgd# &`#$gd{q !!!!!!+!,!-!.!!!!ᴨusdYQhE CJ(aJ(hhE CJ4aJ4hChE CJ0aJ0mH sH UhZhE CJ0aJ0mH sH )jhZhE CJ0EHUaJ0mH sH j_B hE CJUVaJhE CJ0aJ0mH sH jhE CJ0UaJ0mH sH hE 5CJ4aJ4mH sH hbhE 5CJ4aJ4mH sH hE hbhE CJaJmH sH hE CJaJmH sH 0!k!l!!!¨ĨƨdfhjnpީIJKLnz$a$gd?*gdE The correlation seems strong  but if we calculate it we ll find that r = 0  residual error  EMBED Equation.3 , true value  EMBED Equation.3 , predicted value  The least squares principle:  EMBED Equation.3  So what we re lo¨ƨȨ:<>@Hdjnpԩ;աwfաVjhE CJ(UaJ(mH sH !jhLohE EHUmH sH jB hE CJUVaJhLohE CJ(aJ(mH sH hOXCJ(aJ(mH sH hE CJ(aJ(mH sH !jh*hE EHUmH sH j8B hE CJUVaJhE mH sH jhE UmH sH hE hhE CJ(aJ(hOXCJ(aJ(h/CJ(aJ(ԩ֩ةکHILV]mnuyz{˿ti]hCJ0aJ0mH sH hhCJ$aJ$hhCJ$aJ$mH sH hyhyCJ0aJ0mH sH h?*CJ0aJ0mH sH hyCJ0aJ0mH sH hE hAhE CJ(aJ(mH sH UhE CJ(aJ(mH sH jhE CJ(UaJ(mH sH )jhAhE CJ(EHUaJ(mH sH jÐB hE CJUVaJoking for is the parameters (a, b) of the regression line. 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