# How can I use the value of $R^2$ to test the linearity assumption in multiple regression analysis?

The below graphs are residual scatter plots of a regression test for which "normality", "homoscedasticity" and "independence" assumptions have already been met for sure! For testing the "linearity" assumption, although, by looking at the graphs, it can be guessed that the relationship is curvilinear, but the question is: How can the value for "R2 Linear" be used to test the linearity assumption? What's the acceptable range for the value of "R2 Linear" to decide if the relationship is being linear? What to do when the linearity assumption is not met and transforming the IVs also doesn't help?!!

Here is the link to the full results of the test.

Scatter plots:

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I see by the looks of the graphs that you are using SPSS. Just open the graph to edit and find "Add fit line button" there you find some nonlinear line-drawing options, e.g. Loess. Check if this option gives you a reasonably straight line. – ttnphns Jul 22 '12 at 13:40
@ ttnphns: I added the plot with Loess line 2 the question. – Cyrus Jul 22 '12 at 15:03
Well, it appears quite curvilinear, isn't it? You can play more with Loess parameters to see what happens. If line is curved you can visually conclude the relationship is not linear. – ttnphns Jul 22 '12 at 16:11
@Cyrus, I've posted a general answer to this question but was going to add a bit of interpretation on your plots and realized that I'm not quite sure what the $x$ and $y$ axes are in your plot - can you clarify? – Macro Jul 22 '12 at 17:41
@ ttnphns: yep, it's curvilinear. I dont know how to treat this model! In this test (#2) I have 2 IVs that directly effect the DV (PIT). The regression result showed that only 1 of the IVs significantly effect the DV. The R2 is so low (0.172) & the linearity is also being low (at least, according to the graph, when the IV is at low levels). I dont know if this test is acceptable or not! Even I transformed both IVs (by computing their LN) and re-ran the regression, but the result got even worse! – Cyrus Jul 22 '12 at 18:26
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## 2 Answers

Note that the linearity assumption you're speaking of only says that the conditional mean of $Y_i$ given $X_i$ is a linear function. You cannot use the value of $R^2$ to test this assumption.

This is because $R^2$ is merely the squared correlation between the observed and predicted values and the value of the correlation coefficient does not uniquely determine the relationship between $X$ and $Y$ (linear or otherwise) and both of the following two scenarios are possible:

• High $R^2$ but the linearity assumption is still be wrong in an important way

• Low $R^2$ but the linearity assumption still satisfied

I will discuss each in turn:

(1) High $R^2$ but the linearity assumption is still be wrong in an important way: The trick here is to manipulate the fact that correlation is very sensitive to outliers. Suppose you have predictors $X_1, ..., X_n$ that are generated from a mixture distribution that is standard normal $99\%$ of the time and a point mass at $M$ the other $1\%$ and a response variable that is

$$Y_i = \begin{cases} Z_i & {\rm if \ } X_i \neq M \\ M & {\rm if \ } X_i = M \\ \end{cases}$$

where $Z_i \sim N(\mu,1)$ and $M$ is a positive constant much larger than $\mu$, e.g. $\mu=0, M=10^5$. Then $X_i$ and $Y_i$ will be almost perfectly correlated:

u = runif(1e4)>.99
x = rnorm(1e4)
x[which(u==1)] = 1e5
y = rnorm(1e4)
y[which(x==1e5)] = 1e5
cor(x,y)
[1] 1


despite the fact that the expected value of $Y_i$ given $X_i$ is not linear - in fact it is a discontinuous step function and the expected value of $Y_i$ doesn't even depend on $X_i$ except when $X_i = M$.

(2) Low $R^2$ but the linearity assumption still satisfied: The trick here is to make the amount of "noise" around the linear trend large. Suppose you have a predictor $X_i$ and response $Y_i$ and the model

$$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$$

was the correct model. Therefore, the conditional mean of $Y_i$ given $X_i$ is a linear function of $X_i$, so the linearity assumption is satisfied. If ${\rm var}(\varepsilon_i) = \sigma^2$ is large relative to $\beta_1$ then $R^2$ will be small. For example,

x = rnorm(200)
y = 1 + 2*x + rnorm(200,sd=5)
cor(x,y)^2
[1] 0.1125698


Therefore, assessing the linearity assumption is not a matter of seeing whether $R^2$ lies within some tolerable range, but it is more a matter of examining scatter plots between the predictors/predicted values and the response and making a (perhaps subjective) decision.

Re: What to do when the linearity assumption is not met and transforming the IVs also doesn't help?!!

When non-linearity is an issue, it may be helpful to look at plots of the residuals vs. each predictor - if there is any noticeable pattern, this can indicate non-linearity in that predictor. For example, if this plot reveals a "bowl-shaped" relationship between the residuals and the predictor, this may indicate a missing quadratic term in that predictor. Other patterns may indicate a different functional form. In some cases, it may be that you haven't tried to right transformation or that the true model isn't linear in any transformed version of the variables (although it may be possible to find a reasonable approximation).

Regarding your example: Based on the predicted vs. actual plots (1st and 3rd plots in the original post) for the two different dependent variables, it seems to me that the linearity assumption is tenable for both cases. In the first plot, it looks like there may be some heteroskedasticity, but the relationship between the two does look pretty linear. In the second plot, the relationship looks linear, but the strength of the relationship is rather weak, as indicated by the large scatter around the line (i.e. the large error variance) - this is why you're seeing a low $R^2$.

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Thank you Macro for your detailed answer:) – Cyrus Jul 23 '12 at 8:42

Certainly fitting a smoother like LOESS and seeing how close to linear the fit is, is one way to assess linearity of the function. I want to address the main point of the question which is the extent to which R square can measure linearity. Clearly since an $R^2=1$ means the data fall perfectly on a line. But the question of how close to $1$ does $R^2$ need to be to determine that the curve is linear is more difficult than it may sound. Certainly sample size is a factor. If you have just 3 to 6 points $R^2$ will likely be very high regardless of the shape of the function that may represent the data. Even in large samples the region in which the data is collected matters. Nonlinear functions will look linear locally. This is particularly true for polynomials. Consider the function y=x$^2$. In the region $1<x<2$ the function looks linear and data generated from this model with a little additive noise will lead to a high value for $R^2$. On the other hand the model could be perfectly linear but have a large noise component and $R^2$ could be small.

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Thanks Michael. My sample size is 302. I would appreciate it if u could have a look at the results of the test here and see if it's plausible and tenable to report. TQ – Cyrus Jul 22 '12 at 19:23
@Cyrus This is a tough one. Residuals look like they fit the normal really well and there is nothing I can see that would be wrong with the linear regression. You have a decent amount of data. R square is low because the random noise component is large. The LOESS plot shows some curvature at the lower values of the independent variable. But I don't find that convincing. I think it well could be linear and it show why R square is not a good indicator in this case. – Michael Chernick Jul 22 '12 at 20:04
Tq Michael:) Yes, it's really perplexing! All the assumptions are perfectly met but linearity! As u can see in the 1st graph above, the quadratic R2(0.199) is bigger than the linear R2(0.172) which means that it can predict the model better. Actually when i did quadratic regression (by adding SC2) the scatter plot in the result was so heteroscedatic! I'm so confused! Dont know what to do with this model! It's only problem is its low linearity. I dont know how to justify the linearity if I put the scatter plot in my report. Quadratic regression also fails 2meet the homogeneity assumption. Help – Cyrus Jul 22 '12 at 22:12
I don't think it is perplexing. It looks fairly linear. There is a lot of variability which is why R square is low. I think the only way to you could reduce the variability would be to find anothe explanatory variable. – Michael Chernick Jul 22 '12 at 22:45