For example, we might test a drug by administering it in doses that vary from zero to some maximum value. In contrast, in a linear regression we assume that the experimental factor has a varying effect that increases linearly with the size of the factor. That effect manifests itself by creating a difference in the mean value of the measured quantity. In a simple one-factor experiment such as the blood pressure study example of Sections 4.2 through 5.3, we assume that the presence of the factor has a fixed effect on the size of the measured quantity in the trials. In this case, we additionally state that the factor graphed on the horizontal axis has a significant effect on the factor graphed on the Y axis. If the deviations from a slope of zero are great enough for the value of P to fall below 0.05, then we say that the best fit line differs significantly from zero. In this particular test, P states the probability that a slope as great or greater than that of the best fit line would occur due to chance deviations from an actual slope of zero. Recall the general interpretation of P: if the null hypothesis were true, P is the probability that deviations as great as or greater than those seen in the sample would occur due to chance unrepresentative sampling. When we conduct the regression statistical test, the software will calculate a value of P. In a regression, our null hypothesis is that the slope of the best fit line is zero. We can frame this situation using the same language of Section 5.2. Because there were only five data points, this would be a fairly likely outcome. However, one could make the case that tail-waging rate is unrelated to treat size and that it was simply a coincidence that the points on the right side of the graph were higher than those on the left side. In the example to demonstrate how to create a scatter plot in Excel, we saw that the best-fit line through the treat size/tail-wagging rate data had a positive slope. A linear regression can facilitate both of these uses. If the tail wagging rate were completely random and not affected by the size of the treat, the best-fit line would be horizontal, showing that the average wag rate was the same regardless of the treat size. In that case, what we want to know is whether the slope of the best fit line is different from zero or not. We may not actually care about describing the way that the factors vary (the equation) we may just want to know if they vary significantly or not. In that case, we simply want to know if changing the size of the treat has a significant effect on the rate of tail wagging. The other important use of linear regression is as a statistical test of significance. If we know the equation of the best-fit line we can plug numbers into it to calculate the predicted value. In the example, we might like to predict how fast the tail will be wagged given a treat which is a size that we didn't specifically measure. One important use of linear regression is predictive. When we fit the best line through the points of a scatter plot, we usually have one of two goals in mind. (There is another chart type that Excel makes which does that - don't use it!) The name of the process used to create the best-fit line is called linear regression. In scientific graphs, one almost never "connects the dots". That's why another name for trendline is best-fit line. It may or may not actually pass through any particular points. You should notice that the trendline is the best line that fits through the points. In the example above, we had Excel calculate and plot a linear trendline through the points.
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