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I am confused about the rule of the slope in the simple linear regression model. I do understand that the slope indicate the rate of change in the response variable, Y, when the explanatory variable, X, is increase by one unit. For example,

Suppose I have the following simple linear regression equation, where Y is the fat gain in kelogram and X is the non-excercise activity for adult people.

y = 3.505 - 0.00344 X

So to interpret the slope, I would say, foe one unit increase in X, then the fat gain will decrease by 0.00344. The slope is small, hence X does not affect Y.

Is my interpretation correct?

I read this: " the small value of the slope does not mean that the affect of the increased non-excercise activity on fat gain is small - it just reflect the choice of the kilogram as the unit for the fat gain. The slope and intercept of least square line depend on the units of measurement- you cannot conclude anything from their size".

This makes me confuse.

Any help please.

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    $\begingroup$ I agree with you in that the reason given in the answer, "reflecting the choice of unit of y", can be confusing. If $y$ was given in gram, then the equation would be $y=3505 - 3.44 x$. Compared to the intercept, the slope is still small, but not absolutely - that is probably what the author(s) wanted to say. $\endgroup$
    – Ute
    Jun 5, 2023 at 23:19

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In a simple regression problem, the slope describes an average increase of y when x increases by 1 unit. What you are showing here is just an equation: y = f(x). Your question seems more about understanding the slope of a line in general.

Your interpretation assumes that a small slope indicates a small effect because it is small. However, the effect of a treatment can be a large multiple of your slope. It does not have to be a unit increase.

Imagine that x is the amount of seconds spent in training during 10 days and that you train 2 hours per day. Let x=0 before you start training. After training x=10 * 2 * 60 * 60=72 000.

Hence, the effect of that short training program on y is 72 000 * (- 0.00344) = -247.68.

Now, let's turn back to the kilograms. Let y be the fat gain in kg. Say, we are working with a microscopic animal, weighting 10mg.

To get an effect relevant for that animal, we convert the effect to be at the mg scale. A decrease of 0.00344kg corresponds to a decrease of 3440.0mg per unit increase in x. Is that a small effect?

In any case, our small animal may soon go extinct.

Hope this help a bit.

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