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I am just learning to understand how to utilize residual plots to improve regression models. It seems like the following residual vs. fitted plot is hugely problematic as it has a pattern. But how do I go about identifying the most likely causes for such an issue?

The following is a residual vs fitted plot when the dependent variable is log-transformed. I guess the same issue of pattern persists:

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  • 2
    $\begingroup$ Can you elaborate more on the model that generated this residual plot? What was the dependent variable? What were the independent variables? Did the model include any interactions between any of the independent variables? We can't really comment on improvements to your model if we don't even know what the model is. Also, what is the final purpose of your model? $\endgroup$ Apr 11, 2018 at 16:37
  • 3
    $\begingroup$ Thank you @IsabellaGhement. The dependent variable is the number of days it took for an individual to get a loan. The predictors in the model include requested loan size (in $), the number of borrowers, and loan description (text analytic measures as relative frequency counts of words; e.g., positive tone/negative tone). The model does not include any interactions. I hope this provides some clarifications. $\endgroup$
    – SanMelkote
    Apr 11, 2018 at 16:45

1 Answer 1


A "good" residuals vs fitted plot for a linear model might look something like this

df <- data.frame(e = rnorm(2000, 0, sd = 1.5),
                 yhat = runif(2000))
ggplot(df, aes(x = yhat, y = e)) +
  geom_abline(intercept =  0, slope = 0) +
  geom_point() +
  geom_smooth() +
  labs(y = "Residuals", x = "Fitted")

enter image description here


  • how the residuals are spread evenly around 0 throughout the range of the fitted value,
  • the residuals have the same variance --- they are evenly spread the ~ the same distance either side of zero throughout the range of the fitted values (i.e. there's about the same number of residuals $>|3|$ for example at each location on the x-axis.)
  • there is no strong systematic pattern in the residuals; the blue line is similar to the red one in your plot and is a scatterplot smoother showing pattern in the mean of residuals.

In your plot we notice two signifant problems:

  1. There is clear non-constant variance. The spread of the residuals towards the left of the plot is much less than the spread towards the right of the plot.
  2. There is substantial bias in the estimated values; all the residuals are negative above a fitted value of ~40. Similarly, all the residuals at small fitted values (towards the left of the plot) are positive.

These suggest that

  1. the conditional distribution of the response is not Gaussian. In other words, once you account for the effects on the response of your predictor variables (covariates), the response is not normally distributed.

From your reply to @Isabella, an immediate problem is that the response variable is number of days to get a loan. This is unlikely to be conditionally distributed Gaussian because you can't have a negative number of days; the distribution of the response is bounded at 0. Typically, such count data have a non-constant mean-variance relationship; the variance (spread of data about the mean) increase as the mean value increases.

Counts like this are typically modelled as being conditionally distributed Poisson, using a Generalised Linear Model (GLM) with a Poisson family and log link. This is often a useful starting point but other distributions are also used for integer counts that can account for more complexity in the observed counts than can be accounted for in the Poisson.

If your response is not an integer number of days (2.4 days is an allowed value) then you would need a distribution that has support on the positive real values (any value > 0). Such distributions include the Gamma distribution, which also can be fitted as a GLM.

If you do not have an 0 day values in the data set, then you could quickly try to fix the problem of the wrongly assumed mean-variance relationship by applying a log-transformation to the response variable (lm(log(y) ~ x1 + x2) in pseudo code) and see if that improves things. But typically one would prefer a GLM-like model so that one can estimate the mean of the response directly rather than the mean of the log of the response.

From the limited amount of information you have provided, there is little more than I can say beyond the above.

  • $\begingroup$ Thanks @GavinSimpson for such detailed and very helpful clarifications and suggestions. Based on your input, I first attempted the log-transformation of the dependent variable (days-to-fund-loan). The residual vs fitted plot shows an even spread until fitted value < 8, and then a clear (somewhat steep) downward slope pattern. As I understand, this is problematic as well. I will try GLM with Poisson family. Are similar residual diagnostic plots and interpretations applicable for Poisson family? $\endgroup$
    – SanMelkote
    Apr 11, 2018 at 17:44
  • $\begingroup$ To your other point @GavinSimpson, I have the flexibility of coding the days-to-fund the loan as either 2.4 days or round to 2 days, and similarly code the same day funded loans as either 0.5 or 0 days. As you suggested, I will check out Gamma distribution as well. $\endgroup$
    – SanMelkote
    Apr 11, 2018 at 17:45
  • $\begingroup$ You can't just reply on or use this one diagnostic plot to identify all modelling issues. Your model clear still has considerable bias at larger fitted values. Look at the fitted versus observed values and you'll see the bias problem. Also, you can plot residuals against your predictor variables to diagnose if you need different functional forms for the effects of your covariates. $\endgroup$ Apr 11, 2018 at 18:01

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