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# L5h - Bivariate data Looking for a linear relationship The...

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Bivariate data Looking for a linear relationship The least-squares line Using R Checking residuals MATH1005 Statistics Lecture 5 M. Stewart School of Mathematics and Statistics University of Sydney

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Bivariate data Looking for a linear relationship The least-squares line Using R Checking residuals Outline Bivariate data Looking for a linear relationship The least-squares line Using R Checking residuals
Bivariate data Looking for a linear relationship The least-squares line Using R Checking residuals Bivariate data Until now we have only considered univariate data , where each observation is a single number. We also come across situations where for each “individual”, two or more numbers are recorded: multivariate: more than one number; bivariate: exactly two numbers. For bivariate data we usually have an independent variable ( x ) and a dependent variable ( y ). Interest centres on how y behaves in relation to x . Rarely do the y -values behave as a simple function of x . Usually the best we can do is to say that the y -values behave approximately like a simple function of x . What does this mean?

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Bivariate data Looking for a linear relationship The least-squares line Using R Checking residuals Decompose data Non-random part + random part We can perhaps decompose each y -value into a non-random part and a random part. More precisely, for each i = 1 , 2 , . . . , n we can write y i = f ( x i ) + e i where f ( · ) is some (simple) function and the resultant residuals or errors e 1 , e 2 , . . . , e n are randomly distributed about zero. The value f ( x i ) represents the “systematic” component of y i , dependent on x i while the residual e i (= y i - f ( x i )) represents the “noise”, should have no dependence on x i .
Bivariate data Looking for a linear relationship The least-squares line Using R Checking residuals What sort of function f ( · )? Is the relationship (i.e. the function f ( · )) monotone: do larger values of y tend to be associated with larger (smaller?) values of x ?

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