The test statistic for a linear regression is ts=/. It gets larger as the degrees of freedom (n−2) get larger or the r2 gets larger. Under the null hypothesis, the test statistic is t-distributed with n−2 degrees of freedom. When reporting the results of a linear regression, most people just give the r2 and degrees of freedom, not the ts value. Anyone who really needs the ts value can calculate it from the r2 and degrees of freedom.
Linear regression and correlation assume that the data points are of each other, meaning that the value of one data point does not depend on the value of any other data point. The most common violation of this assumption in regression and correlation is in time series data, where some Y variable has been measured at different times. For example, biologists have counted the number of moose on Isle Royale, a large island in Lake Superior, every year. Moose live a long time, so the number of moose in one year is not independent of the number of moose in the previous year, it is highly dependent on it; if the number of moose in one year is high, the number in the next year will probably be pretty high, and if the number of moose is low one year, the number will probably be low the next year as well. This kind of non-independence, or "autocorrelation," can give you a "significant" regression or correlation much more often than 5% of the time, even when the null hypothesis of no relationship between time and Y is true. If both X and Y are time series—for example, you analyze the number of wolves and the number of moose on Isle Royale—you can also get a "significant" relationship between them much too often.
Keywords: comparative correlative, left-subordinate hypothesis, ..
While reduced major axis regression gives a line that is in some ways a better description of the symmetrical relationship between two variables (McArdle 2003, Smith 2009), you should keep two things in mind. One is that you shouldn't use the reduced major axis line for predicting values of X from Y, or Y from X; you should still use least-squares regression for prediction. The other thing to know is that you cannot test the null hypothesis that the slope of the reduced major axis line is zero, because it is mathematically impossible to have a reduced major axis slope that is exactly zero. Even if your graph shows a reduced major axis line, your P value is the test of the null that the least-square regression line has a slope of zero.
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If you are mainly interested in using the P value for hypothesis testing, to see whether there is a relationship between the two variables, it doesn't matter whether you call the statistical test a regression or correlation. If you are interested in comparing the strength of the relationship (r2) to the strength of other relationships, you are doing a correlation and should design your experiment so that you measure X and Y on a random sample of individuals. If you determine the X values before you do the experiment, you are doing a regression and shouldn't interpret the r2 as an estimate of something general about the population you've observed.
Inverse gambler's fallacy - Wikipedia
In our enhanced Pearson’s correlation guide, we also show you how to write up the results from your assumptions tests and Pearson’s correlation output if you need to report this in a dissertation, thesis, assignment or research report. We do this using the Harvard and APA styles. We also show you how to write up your results if you have performed multiple Pearson’s correlations. You can learn more about our enhanced content .
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For the most part, I'll treat correlation and linear regression as different aspects of a single analysis, and you can consider correlation/linear regression to be a single statistical test. Be aware that my approach is probably different from what you'll see elsewhere.