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I have the following r code. I created a multiple linear regression model on a math_and_alcohol dataset. I can see in the summary of the model that the r-squared is .8279 which means the model explains 83% of the variability around the mean.

Here is a list of most of the predictors. They include things like gender, age, family size, mother's/father's educational status and job status, family and school support, travel distance, free time, how much time student spends going out, drinking during the week and drinking on the weekend, internet access, health, etc

  ..$ sex       : Factor w/ 2 levels "F","M": 1 1 1 1 1 2 2 1 2 2 ...
  ..$ age       : int [1:395] 18 17 15 15 16 16 16 17 15 15 ...
  ..$ famsize   : Factor w/ 2 levels "GT3","LE3": 1 1 2 1 1 2 2 1 2 1 ...
  ..$ Pstatus   : Factor w/ 2 levels "A","T": 1 2 2 2 2 2 2 1 1 2 ...
  ..$ Medu      : int [1:395] 4 1 1 4 3 4 2 4 3 3 ...
  ..$ Fedu      : int [1:395] 4 1 1 2 3 3 2 4 2 4 ...
  ..$ Mjob      : Factor w/ 5 levels "at_home","health",..: 1 1 1 2 3 4 3 3 4 3 ...
  ..$ Fjob      : Factor w/ 5 levels "at_home","health",..: 5 3 3 4 3 3 3 5 3 3 ...
  ..$ traveltime: int [1:395] 2 1 1 1 1 1 1 2 1 1 ...
  ..$ studytime : int [1:395] 2 2 2 3 2 2 2 2 2 2 ...
  ..$ failures  : int [1:395] 0 0 3 0 0 0 0 0 0 0 ...
  ..$ schoolsup : Factor w/ 2 levels "no","yes": 2 1 2 1 1 1 1 2 1 1 ...
  ..$ famsup    : Factor w/ 2 levels "no","yes": 1 2 1 2 2 2 1 2 2 2 ...
  ..$ activities: Factor w/ 2 levels "no","yes": 1 1 1 2 1 2 1 1 1 2 ...
  ..$ nursery   : Factor w/ 2 levels "no","yes": 2 1 2 2 2 2 2 2 2 2 ...
  ..$ higher    : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
  ..$ internet  : Factor w/ 2 levels "no","yes": 1 2 2 2 1 2 2 1 2 2 ...
  ..$ romantic  : Factor w/ 2 levels "no","yes": 1 1 1 2 1 1 1 1 1 1 ...
  ..$ famrel    : int [1:395] 4 5 4 3 4 5 4 4 4 5 ...
  ..$ freetime  : int [1:395] 3 3 3 2 3 4 4 1 2 5 ...
  ..$ goout     : int [1:395] 4 3 2 2 2 2 4 4 2 1 ...
  ..$ Dalc      : int [1:395] 1 1 2 1 1 1 1 1 1 1 ...
  ..$ Walc      : int [1:395] 1 1 3 1 2 2 1 1 1 1 ...
  ..$ health    : int [1:395] 3 3 3 5 5 5 3 1 1 5 ...
  ..$ absences  : int [1:395] 6 4 10 2 4 10 0 6 0 0 ...

math_grade_model <- lm(G3 ~ ., math_and_alcohol)
summary(math_grade_model)
summary(math_grade_model)$r.squared

I now want to assess if this linear model is a good model. So I plotted the fitted values against the residuals. I also made a histogram of the residuals and I made a qq plot. I've been doing some reading and these are suggestions I have found.

My problem is I don't have any idea how to read these graphs!

The histogram looks a little left-skewed to me but mostly normal I think? The qq plot looks like a straight line except there's a serious tail on the left side. I have no idea how to understand the fitted values versus the residuals.

If anyone could help I would really appreciate it!

ggplot(data = math_grade_model, aes(x = .fitted, y = .resid)) +
    geom_point() +
    geom_hline(yintercept = 0, linetype = "dashed") +
    xlab("Fitted values") +
    ylab("Residuals") +
    labs(title = "Math Grade Model", subtitle = "Scatterplot - Math Grade  Linear Model") +
  theme(plot.title = element_text(hjust = 0.5, face = "bold")) +
  theme(plot.subtitle = element_text(hjust = 0.5)) +
  labs(caption = "HW")

ggplot(data = math_grade_model, aes(x = .resid)) +
  geom_histogram() +
  xlab("Residuals") +
  labs(title = "Math Grade Model", subtitle = "Histogram - Math Grade Linear Model") +
  theme(plot.title = element_text(hjust = 0.5, face = "bold")) +
  theme(plot.subtitle = element_text(hjust = 0.5)) +
  labs(caption = "HW")

ggplot(data = math_grade_model, aes(sample = .resid)) +
  stat_qq() + stat_qq_line() +
  labs(title = "Math Grade Model", subtitle = "QQ Plot - Math Grade Linear Model") +
  theme(plot.title = element_text(hjust = 0.5, face = "bold")) +
  theme(plot.subtitle = element_text(hjust = 0.5)) +
  labs(caption = "HW")

enter image description here

enter image description here

enter image description here

@Nick Cox I added another graph as you suggested. Now I've encountered a little bit of a coding question -- I think what I'm doing here is putting G3 on the x axis and then the fitted values from the model on the y-axis. I've never created this kind of graph before. Is this what you were picturing?

ggplot(data = math_grade_model, aes(x = math_and_alcohol$G3, y = .fitted)) +
    geom_point() +
    geom_hline(yintercept = 0, linetype = "dashed") +
    xlab("Observed Values") +
    ylab("Fitted Values") +
    labs(title = "Math Grade Model", subtitle = "Scatterplot - Math Grade  Linear Model") +
  theme(plot.title = element_text(hjust = 0.5, face = "bold")) +
  theme(plot.subtitle = element_text(hjust = 0.5)) +
  labs(caption = "HW")

enter image description here

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  • $\begingroup$ You should say a little more about your data and type of predictors. $\endgroup$
    – user293815
    Commented Nov 18, 2020 at 8:39
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    $\begingroup$ The identity residual $\equiv$ observed $-$ fitted means that each distinct observed grade (e.g. 42) defines a line such as 42 $-$ fitted. So the pattern of the residual versus fitted plot is, as seen, one of parallel lines, all with slope $-$1. You don't tell us but let's assume that in principle the grade cannot be negative. Hence the line 0 $-$ fitted is a bound to the plot. Otherwise put, your observed grade can't be negative but a linear model is not so constrained. A plot of observed versus fitted can be a little easier to think about than a residual versus fitted plot at this stage. $\endgroup$
    – Nick Cox
    Commented Nov 18, 2020 at 10:39
  • $\begingroup$ Hi @erc I added some information about the data and type of predictors, as you suggest $\endgroup$
    – hachiko
    Commented Nov 18, 2020 at 21:15
  • $\begingroup$ @Nick Cox I added another graph as you suggested.I hope I did that correctly. I'm putting the observed values on the x and the fitted values on the right. Now it makes a staircase going up to the right? $\endgroup$
    – hachiko
    Commented Nov 18, 2020 at 21:21
  • 1
    $\begingroup$ There are different interpretations of versus Mine is at stats.stackexchange.com/questions/146533/… No matter; which variable goes on which axis is a matter of convention or taste. To the point, your plot shows unsurprising difficulty in predicting zero grades, as some people have negative predictions. Less obviously, your model just occasionally predicts more than 20 when (it seems) top marks are 20. These are flaws of principle but otherwise your model seems to fit well -- but with so many predictors that is unsurprising. $\endgroup$
    – Nick Cox
    Commented Nov 18, 2020 at 21:34

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