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I have a simple question about QQ plots in simple linear regression, but I am a bit confused about the plot: depending where i look the y axis is different: it can be either 1) residuals 2) standardized residuals 3) the actual independent variable. I wonder which one is better to use? I know that we are checking the assumption of normality: that errors are normally distributed (so residuals ). Does it imply that my independent variable is also normally distributed? and what should be a proper y-axis?

enter image description here enter image description here enter image description here

I got a response in one of the chat: They are all identical for the purpose of normality checking. If the residual, epsilon, is normally distributed then Y = XB + epsilon is normally distributed. .... But the technical condition necessary for inference (p-values) is that the residuals are normally distributed. This is called called the conditional distribution of Y, conditioned on all the X's.

I am not exactly sure I understand the explanation (except the part about residuals and standardized residuals)

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  • $\begingroup$ Is your question about how to interpret a Q-Q plot, which type of Q-Q plot you should use (i.e., what variable to include on the y-axis), or which assumption of normality is required for exact inference in linear regression? Please clarify, since the title, text, and final question seem to be about different things. $\endgroup$
    – Noah
    Commented May 11, 2022 at 2:38
  • $\begingroup$ i think both. My first question is which type of Q-Q plot you should use? and my second question is basically related to the first one: if we (for example do not do Y variable on y axis) I wonder why $\endgroup$
    – yuliaUU
    Commented May 11, 2022 at 2:51
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    $\begingroup$ @BruceET, just to make sure i got it write: you are saying that putting either standardized residuals or the just residuals is OK ( or you also mean that I can also put my Y variable on y-axis). cause the title "sample" does not really tell me much whether we use residuals or actual y-values $\endgroup$
    – yuliaUU
    Commented May 11, 2022 at 3:48
  • $\begingroup$ Sorry for delay. Had to leave the computer for a while. Either method is correct. I have deleted by comment above because I have expanded into a formal Answer, with graphs. // "Sample" refers to data. "Theoretical: refers to population CDFs $\endgroup$
    – BruceET
    Commented May 11, 2022 at 5:03

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Both styles of Q-Q plots are considered correct and both are in common use. The default in R is to put the 'data' or 'sample' quantiles on the vertical axis; parameter datax=T puts the theoretical quantiles on the vertical axis. . If you're showing the Q-Q plot along with a histogram or an ECDF plot, the latter style seems natural.

set.seed(2022)
x = rnorm(100, 100, 15)
par(mfrow=c(1,3))
 hist(x, prob=T, col="skyblue")
  curve(dnorm(x, 100, 15), add=T, col="blue", lwd=2)
 plot(ecdf(x))
  curve(pnorm(x, 100, 15), add=T, col="blue", lwd=2)
 qqnorm(x, datax=T)
  qqline(x, datax=T, col="blue", lwd=2)
 par(mfrow=c(1,1))

enter image description here

However, if you want to draw a line around which points of a normal sample might fall, that's easier to do with the default style of Q-Q plot: $y=μ+σx.$ The default reference line in R (with qqline) connects sample and theoretical quartiles.

This default method in R seems to be used more often outside of North America. However, you should feel free to use whichever style of Q-Q plot you prefer. (The only exception might be if you're submitting to a journal that insists that one of the styles must always be used.)

In the figure below, (left) the line $y=\mu + \sigma x = 100 + 15x$ is shown in brown. The blue reference line (right) connects quartiles (red).

par(mfrow=c(1,2))
 qqnorm(x)
  abline(a = 100, b = 15, col="brown")
 qqnorm(x)
  qqline(x, col="blue")
   abline(h = quantile(x, c(.25,.75)), col="red")
   abline(v = qnorm(c(.25,.75)), col="red")
par(mfrow=c(1,1))

enter image description here

Addendum: Mentioned Shapiro-Wilk test in Comment. Results of S-W test for sample x used above are shown below. Sample would be considered consistent with sampling from some normal population because the P-value of the test exceeds 5%.

shapiro.test(x)

        Shapiro-Wilk normality test

data:  x
W = 0.99017, p-value = 0.678
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  • $\begingroup$ why both are acceptable? i thought we testing assumption that errors are normally distributed, not a response variable? $\endgroup$
    – yuliaUU
    Commented May 11, 2022 at 6:24
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    $\begingroup$ The two styles show the same information. Most people get familiar with one of the two for personal convenience. // Strictly speaking, neither provides a formal test of normality, but Q-Q plots are often used informally to judge normality, especially for small samples where formal tests may not have good power. $\endgroup$
    – BruceET
    Commented May 11, 2022 at 14:08
  • $\begingroup$ in your example, x represent the dependent or independent variable? $\endgroup$
    – yuliaUU
    Commented May 11, 2022 at 17:14
  • $\begingroup$ In a Q-Q plot there is only one variable 'x' and we wonder whether it was randomly sampled from a normal distribution. For a sample of size $n=100,$ one might use the Shapiro-Wilk normality test $(H_0: Normal).$ But for various reasons, the S-W test may not give useful results for very small or very large samples. Many statisticians prefer to look at a Q-Q plot and conclude sample likely normal if points on Q-Q plot fall 'nearly' in a straight line (not being too fussy about a few 'nonlinear' points in the tails of x. // In a regression or ANOVA one expects residuals from model to be normal $\endgroup$
    – BruceET
    Commented May 11, 2022 at 17:23
  • $\begingroup$ Soory, I think I was not very clear in my question. In a context of SLR, lets say in data cars : lm(dist~speed), when i do QQ plot, what is x variable is? am I correct that I can calculate standardized residuals and plot it against the theoretical quantiles (using qqnorm() ) OR I can take the response variable (dist) and plot it against theoretical quantile? $\endgroup$
    – yuliaUU
    Commented May 11, 2022 at 18:13

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