# Choosing a significance test, trend appears either linear or quadratic

I am testing a one-sided hypothesis of association between two variables. I want to know if I increase variable A, then will variable B also increase?

For my experimental design, I increase the variable A in a defined interval and calculate the result (B) at the end of my measurement. By plotting this results B over the variable A, the signal is noisy. A rise upwards is visible but not over the whole range of inputs. I'm not sure if the trendline is a linear or quadratic form. The data are normally distributed. I confirmed that by test with $$\chi^2$$-test.

Now I want to do a significance test to assess my hypothesis. I have done the R2-Test for the linear model. Can I use the R2 also vor squared distributed data? How can report confidence intervals? I also used the F-Test. xI considered different designs, like a factorial design, but this seems not to be possible.

What is the significance test to assess my hypothesis? What test says, that variable A and variable B are dependent or not? Is there a coefficient of determination?

## 1 Answer

1. Statistical tests do not prove hypotheses. Some say that tests can disprove (null) hypotheses, but even that is not right since there is still risk of false-positive findings. It is better to say that a statistical test assesses a hypothesis.

2. You do not test normality of "the data". To assess the assumptions of the (exact finite sample) linear model, you must assess the normality of the residuals. This is an interesting paradox for some: if the wrong model produces normal errors, but a better trendline creates non-normal errors, which do you choose? Well, the scientific question should guide you. Choose the model that best answers the question and deal with both the trend and the residuals in robust ways. The OLS needs no adjustment if the residuals are nonnormal because of the Lindeberg-Feller CLT. If there is heteroscedasticity, use sandwich errors if $$n > 40$$, or jackknife error estimates.

3. If you want to know if B increases when A increases, fit the linear model (adjust for A only). This is not the most powerful modeling approach, but the result is clear inference about the direction and magnitude of trend over the range of inputs (A) to the model.

4. If you want a powerful association test between A and B fit the linear and the quadratic (A and A^2 as predictors) terms in the model. Do a two-degree-of-freedom test against the null hypothesis which is an intercept only model.

5. In either case, fitting and interpreting the quadratic term is an excellent way to assess the sensitivity of the main findings. The statistical significance of a quadratic term is typically taken to indicate a departure from linearity. The quadratic term locally conforms the trendline to a whole range of other possible functional forms like logarithmic, exponential, or even sinusoidal over some domains. It is less powerful to detect S-curves. Importantly, the significance of the quadratic term does not mean the trend is quadratic, that is an incidental finding. The hypothesis was informed by your inspecting the scatter plot, so you cannot use the same data to generate hypotheses and then test them.

• Dear @AdamO, thanks a lot for your help. I studied your answer, and did some of your explanation you wrote down. 1. Can I use the R2 Test also for not linear regressions? For exmaple a Sigmoid function? 2. Which test do you mean with "two-degree-of-freedom"? I didn't found a specific example for my problem. – joe Jun 22 at 18:08
• @joe R^2 is a useful summary measure for non-linear least squares, makes no sense for GLM: logistic regression estimates a sigmoid but don't use R^2 for that.. 2. The degrees of freedom is the number of free parameters between nested models you wish to test. I think this is a matter of applying theory to your problem rather than finding a specially tailored example. – AdamO Jun 24 at 13:31