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UMass Amherst PSYCH 240 - Correlation and Prediction and Probability

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PSYCH 240 1st Edition Lecture 6Current LectureSection 4: Correlation and Prediction cont.Y-intercept (a)-Y-intercept is the predicted Y value when X=0-The Y-intercept wont make sense if ) is not a possible X value. In this case, just think of the intercept value as something you need to move your line up or down to line it up with the data-Fitting a regression function will rarely produce prediction for the criterion variable outside of the range of data that you have for the predictor variable Regression Line-The least-squares regression line minimizes the sum of squared deviations between the predicted and observes Y values-That is, no other line will make better predictions for Y based on X-Functions other than a line might be able to make better predictions, though-If you take the deviation for every point on the plot, square them, and add them all up, then you will get a lower total with this line than with any other line you could draw Strength of Relationship-We need a way to express how strongly the two variables are related-Slope does not give this strength by itself-Slope is affected both by the strength of the relationship AND the relative scale of the two variables-Slope is not a reliable estimate of relationship strength, because it is affected by scale (units of measurement, etc.)-"Proportion of Variance Accounted For" - r2o This statistic measures how much better we can predict Y using information about X versus NOT using information about XoEx.These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.or2= 0: means there is no relationship between the two variables. Knowing about one variable doesn’t help you predict the other variable at alloR2 = 1: means there is a perfect relationship between the two variables. Knowing about one variable allows you to perfectly predict the value of the other variables every time o-CorrelationoA correlation is a measure of the degree AND direction of relationship between two variablesoFor a correlation, we do a regular regression EXCEPT that we convert both variables to z-scores firstoSo a correlation is the slope you get from a regression on z-scores-You can get r2 by simply squaring the correlation coefficient ® Interpreting r-Sign (+ or -) indicates direction of the relationshipo(+) means Y increases as X increaseso(-) means Y decreases as X increases-The absolute vale indicated the strength of correlationo0 is weakest correlation (no relationship)o1 or -1 is strongest correlation (perfect relationship) Linearity of Relationship-The correlation coefficient assumes a linear relationship between two variables-If the variables are related by aby other type of function, then the correlation coefficient might be low even though the variables are strongly related-You should always look at scatterplots to make sure the relationship between the variables islinear. If it isn't, then don't use linear regression or correlation Linearity-Non-linear relationship: relationship between variables approximately follows pattern that is not a straight line Causality-Three possibilities/directions of causalityoX could be causing YoY could be causing XoAnother variable could be causing both X and Y-Correlation does not imply causation but all correlational evidence cannot be dismissed automatically. You have to evaluate each case to figure out which pattern of causation is most likely Establishing Causality-What do we do when we want to make causal statements?oRun an experiment-Instead of just measuring two variables and trying to predict one with the other, we directly manipulate and independent variable (IV) and measure its effects on a dependent variable (DV)-All variables except the IV are either held constant across conditions or randomly assigned to conditions or randomly assigned to conditions to prevent a systematic effect. Any changes in the DV should be uniquely to the IVoGet more information on the time course of the relationship and possible outside variables that might produce the relationship-If you can rule out plausible outside variables and establish that changes in X precede changes in Y, then you have stronger evidence that X causes YoFigure out HOW one variable causes the other-We can develop theories of the mechanisms by which one variable changes another, and test these theories in controlled experiments. With knowledge of the underlying mechanisms, we can make stronger inferences about relationships we find in the "real" worldObjectives--- Be able to define and recognize examples of a scatterplot. --- Know what it means to say that a line is the Least-Squares Regression Line; that is, what property makes a line the least-squares regression line. --Be able to recognize and describe the components of the linear equation (Y-hat, X, a, and b). Given a linear equation, be able to compute the predicted Y value for a given X value. --Know the definitions for “predictor variable” and “criterion variable,” and know which is which in the linear equation. --Know the definitions for intercept and slope, and be able to recognize these components in the linear equation. --Be able to find the slope of a linear function given two X values and the corresponding predictedY values. --Be able to recognize the difference between functions with positive or negative slopes. Know how to interpret the sign on a slope coefficient (that is, what do positive and negative slopes mean?). --Understand why the regression slope does not accurately measure the strength of the relationshipbetween two variables. --Be able to describe what “proportion of variance accounted for” means and know that this statistic is labeled r2. Understand how r2 relates to the strength of relationship (e.g., given two r2 values, be able to say which indicates a stronger relationship). Be able to match plots with r2 values.--Know what a correlation is, and know that it is equal to the slope coefficient that you get if you convert both variables to z-scores before finding the least-squares regression line. Know that the correlation coefficient is denoted by “r”. --Be able to recognize the difference between plots that show positive and negative correlations.--Know how the value of the correlation coefficient relates to the strength of the relationship between the two variables. Know what values are possible for the correlation coefficient. --- Be able to match


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