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In an experimental physics activity, I registered 5 values corresponding to 5 tests. enter image description here The second column corresponds to the experimental value. The third column corresponds to the estimated value (the value obtained by linear regression). The fourth column corresponds to residuals. I plotted a linear regression and calculated the residuals and plotted the residuals as you can see in the graphs. enter image description hereenter image description here

However, to prove that my linear regression model is adequate for the data, the residuals have to satisfy 4 conditions:

  • Follow a normal distribution

  • Have a zero average

  • Have constant variance (homoscedasticity)

  • Be independent

I was able to prove that the mean was zero (or very close to zero). To prove the independence of residuals, I did the Durbin-Watson test.

The problem is that the Durbin-Watson table gives the values of dL and dU, from n = 6. And I, in my experience, only have 5 experimental values, that is, n = 5. Can I approximate, using the values of dL and dU for n = 6, in the case of n = 5?

If you can not use this test, how do I prove that the residuals are independent?

I also have another question. To prove that the residues follow a normal distribution, I have to do the normal Q-Q test and the normal P-P test. In the case of the normal Q-Q test gave me this chart:

enter image description here

If a linear regression occurs, we conclude that the residuals have a normal distribution, which is what I want to prove.

This is the first time I make a graph of this type (the graph was made in Matlab) and it is not working for me. I would be glad if you clarified what was wrong, or show me the right chart and teach me how to do it.

Everything I've done so far has been self-taught work and I feel like I'm at a standstill. Without your precious help I can not advance.

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  • $\begingroup$ I made your second plot smaller (I'd have shrunk the others as well but the plot symbols and other information are too small to clearly see in a smaller plot). If you really want it bigger again (I'd advise against it, the huge plot sizes make the gist of your text harder to follow), remove the "m" I added just before the ".png" in "T6zbIm.png" $\endgroup$
    – Glen_b
    May 9, 2017 at 0:14

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  1. You can't prove any of the things you're talking about proving. You may in some circumstances be able to see if they're implausible, or you may be able to see if they seem to be reasonably consistent with the assumption.

  2. These assumptions are only necessary if you're relying on something the assumption is used for (such deriving the distribution of the test statistic in a hypothesis test of a coefficient, say, or in obtaining the end-points of a prediction interval).

  3. The zero-average assumption relates to the errors not the residuals. You can't check that the mean of the errors is 0, since you can't observe them; any such consideration of the suitability of that assumption is purely based on understanding from outside the data (such as theoretical considerations) unless you have some way to calibrate what the "true" model values should be (independent of your data). That the mean of the residuals is 0 is an unavoidable mathematical fact (if you fit an intercept and the regression is unweighted), so there's no point in looking at whether it's true for the residuals.

    Indeed the other assumptions are also about errors rather than the residuals, but you can (to some extent) check them using residuals.

  4. You cannot hope to properly assess normality, constant variance or independence with only 5 observations. The large residual on the center observation (it's the center one in the plot vs frequency and in the table) may not fit with those assumptions (but again, it's largely only an issue if you're doing something that actually requires the assumptions to hold). Note with that large residual that it is many times larger than the $\pm 0.001$ you wrote beside the observation. Either the linear-relationship assumption is wrong or there's something odd with that observation.

  5. your Q-Q plot is not correctly implemented. If it has been done correctly the values in the plot must be monotonic

    This is what a Q-Q plot of your raw residuals would look like:

    Q-Q plot of OP's data

    (you may have the axes the other way around; that's not crucial if it's clear, but I think the fixed quantities belong on the x-axis and the random ones on the y-axis myself. If you plot standardized residuals it will look similar but the axis values will be different)

    I can't tell for sure what you did wrong because you don't say what you did, but one possibility is that you didn't have your residuals sorted in the same order as your expected order statistics.

    You can see that there's an impression of curvature in the plot; this is largely due to the relatively large residual of your center point (which is at the top right of this plot)

Note that a problem with any of the assumptions can make it impossible to assess some of the others -- there's little point trying to look at a Durbin-Watson with 5 observations at the best of times, let alone when there's an outlier in the middle of the series.

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