![]() If the scatter plot reveals non linear relationship, often a suitable transformation can be used to attain linearity. One can construct the scatter plot to confirm this assumption. It finds the slope and the intercept assuming that the relationship between the independent and dependent variable can be best explained by a straight line. Linear regression does not test whether data is linear. Statistically, it is equivalent to testing the null hypothesis that the regression coefficient is zero. A related question is whether the independent variable significantly influences the dependent variable. The closer R2 is to 1, the better is the model and its prediction. All software provides it whenever regression procedure is run. Once a line of regression has been constructed, one can check how good it is (in terms of predictive ability) by examining the coefficient of determination (R2). A similar interpretation can be given for the regression coefficient of X on Y. It represents change in the value of dependent variable (Y) corresponding to unit change in the value of independent variable (X).įor instance if the regression coefficient of Y on X is 0.53 units, it would indicate that Y will increase by 0.53 if X increased by 1 unit. The coefficient of X in the line of regression of Y on X is called the regression coefficient of Y on X. We would then be able to estimate crop yield given rainfall.Ĭareless use of linear regression analysis could mean construction of regression line of X on Y which would demonstrate the laughable scenario that rainfall is dependent on crop yield this would suggest that if you grow really big crops you will be guaranteed a heavy rainfall. Here construction of regression line of Y on X would make sense and would be able to demonstrate the dependence of crop yield on rainfall. Choice of Line of Regressionįor example, consider two variables crop yield (Y) and rainfall (X). Often, only one of these lines make sense.Įxactly which of these will be appropriate for the analysis in hand will depend on labeling of dependent and independent variable in the problem to be analyzed. On the other hand, the line of regression of X on Y is given by X = c dY which is used to predict the unknown value of variable X using the known value of variable Y. As you can see, they will only have the same slope if the variances are equal. 1 Cov(x,y) V ar(y) This is regress x against y. This is used to predict the unknown value of variable Y when value of variable X is known. This may be more technical than what you are looking for, but here is the slope of the regression line for both cases: 1 Cov(x,y) V ar(x) This is regress y against x. The line of regression of Y on X is given by Y = a bX where a and b are unknown constants known as intercept and slope of the equation. In contrast, in the example where $X$ and $Z$ are $\pm1,$ learning $X = 1$ gives you no probabilistic information about $Y.$ It's still $\pm 1$ with 50-50 probability, just like before you knew $X.There are two lines of regression- that of Y on X and X on Y. In the example with normal random variables, knowing $X=1$ tells you that $Y = \pm1,$ which is new information about $Y$ since it could have been any real number. The key is that dependence means knowing one variable tells you something about the the other. However, if you replace $X$ and $Z$ in my example with standard normal variables and set $Y = XZ,$ then $X$ and $Y$ are not independent (although they are uncorrelated). For instance, let $X$ and $Z$ be independent RVs, having value $1$ with probability $1/2$ and $-1$ with probability $1/2.$ Then let $Y = XZ.$ $Y$ depends on $X$ in a functional sense, but it is actually independent of $X$. ![]() However, remember this rule of thumb is not always right. In this case, that rule of thumb is correct and others have given the computations and counterexamples demonstrating that. ![]() Your intuition is right that when one variable is a function of the other, they are usually not independent, (although they may be uncorrelated).
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |