Thread: How to investigate a curvilinear relationship? (power curve?)

1. How to investigate a curvilinear relationship? (power curve?)

Hi there,

Can anyone tell me how to explore this relationship on SPSS using a regression analysis? I have been advised that this relationship shows that positive increases in temperature lead to increasingly larger growth in the number of 'passes', and so a curved line fits best. I have tried transforming the variables and also performing a quadratic polynomial regression. Neither have worked.

I would greatly appreciate any advice you may have!

(Scatterplot is attached!)

2. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by crook
Hi there,

Can anyone tell me how to explore this relationship on SPSS using a regression analysis? I have been advised that this relationship shows that positive increases in temperature lead to increasingly larger growth in the number of 'passes', and so a curved line fits best. I have tried transforming the variables and also performing a quadratic polynomial regression. Neither have worked.

I would greatly appreciate any advice you may have!

(Scatterplot is attached!)
By "transforming variables" what exactly do you mean? Did you create a new variable temperature_squared and include it in the regression along with temperature (and night_length, if I recall)?

3. Re: How to investigate a curvilinear relationship? (power curve?)

Is this an assignment and the prof said polynomial should work? It seems like a pretty small data set.

I would fit a linear predictor, graph line overlaid.
I would then fit y = b0 + b1(x1) + b2(x1^2), and graph with the prediction line overlaid.

Pretty much, make some graphs after fitting it. Let us see this beast.

Also, what does "neither worked" even mean???? The program would not run or are you looking at a fit measure or p-value for coefficient to make this generalization. Provide us with some output as well, please.

4. Re: How to investigate a curvilinear relationship? (power curve?)

Thanks for coming to the rescue again!

I have dropped night length from my investigations for various reasons, and am now focusing on temperature only. Yep - I squared the temperature values and added them into the regression as a second independent variable along with temperature. I've added a screenshot of the SPSS output.

5. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by hlsmith
Is this an assignment and the prof said polynomial should work? It seems like a pretty small data set.

I would fit a linear predictor, graph line overlaid.
I would then fit y = b0 + b1(x1) + b2(x1^2), and graph with the prediction line overlaid.

Pretty much, make some graphs after fitting it. Let us see this beast.

Also, what does "neither worked" even mean???? The program would not run or are you looking at a fit measure or p-value for coefficient to make this generalization. Provide us with some output as well, please.
The bold is what I was getting at (plus the OP's other predictor if I recall from his other post). My suspicion is that a squared term will do the trick.

6. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by crook
Thanks for coming to the rescue again!

I have dropped night length from my investigations for various reasons, and am now focusing on temperature only. Yep - I squared the temperature values and added them into the regression as a second independent variable along with temperature. I've added a screenshot of the SPSS output.
If you don't mind my asking, why did you end up dropping night length? Was it nonsignificant (and then temp significant) after dropping the interaction term?
Adjusted R-squared being negative (and the global F-test being nonsignificanct) is a decent indication that the proposed model isn't worth much (barring any issue that would be causing this).

7. Re: How to investigate a curvilinear relationship? (power curve?)

Night length was dropped essentially because my professor decided it wasn't an interesting avenue to explore. He's more keen on a focused investigation into temperature and bat passes.

To clarify what didn't work - prof suggested that the curvilinear relationship could be linearised with a transformation and that way I could run a linear regression between temperature and passes. I've squared temperature but this does not linearise the relationship. I'm sorry if I'm confusing anyone, us biologists are not the strongest at stats.

If I need to add any other screenshots please let me know and I will do so.

8. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by crook
Night length was dropped essentially because my professor decided it wasn't an interesting avenue to explore. He's more keen on a focused investigation into temperature and bat passes.

To clarify what didn't work - prof suggested that the curvilinear relationship could be linearised with a transformation and that way I could run a linear regression between temperature and passes. I've squared temperature but this does not linearise the relationship. I'm sorry if I'm confusing anyone, us biologists are not the strongest at stats.

If I need to add any other screenshots please let me know and I will do so.
Here is the way I view this:
Do we reasonably believe that night length impacts the number of passes (somewhat indicating how active these bats are)? I think this is a reasonable thing. The more time for a nocturnal animal to move about, the more we would expect it to, all else constant. (Although, a similar argument can be made for temperature. I would check the correlation between night length and temperature.)

If you agree with the above, we should include night length in the model whether we're interested in it or not, because it has a real relationship with the outcome variable.
If we include it, then we can also add temperature, saying, "given that we've accounted for night length, is there anything meaningful contributed to predicting the outcome if we include the temperature?" This, in my opinion, is the better approach (in any analysis), because you're accounting for things that are influential on the dependent variable. Further, it doesn't usually make sense to test these "known" variables that we include based on theory, because we know they're important, and we'd be opening ourselves up to making an error in testing.

If your professor is looking to explore this relationship between passes and temperature, it really only makes sense to do so after you've accounted for known influences on the dependent variable.

9. Re: How to investigate a curvilinear relationship? (power curve?)

There is a strong correlation between temperature and night length. I think he felt that any impact of night length could most likely be attributed to the falling temperatures on those longer nights, meaning it wasn't worth including in further investigation. Further, I've also created nightly activity graphs for 3 week periods, with passes by hour, and this standardises for night length.
What's left now is looking at temperature and passes, but this curved relationship is proving to be a big problem. Unfortunately hlsmith's suggestion is like Dutch to me! How do I perform this action in SPSS?

10. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by crook
There is a strong correlation between temperature and night length. I think he felt that any impact of night length could most likely be attributed to the falling temperatures on those longer nights, meaning it wasn't worth including in further investigation. Further, I've also created nightly activity graphs for 3 week periods, with passes by hour, and this standardises for night length.
What's left now is looking at temperature and passes, but this curved relationship is proving to be a big problem. Unfortunately hlsmith's suggestion is like Dutch to me! How do I perform this action in SPSS?
Are you saying that your depending variable is passes per hour or just total passes in the night?

As for the bold, in other words, does he believe that "after we account for the effect of temperature, night length has no influence on total bat activity (more or less)"?
To me, it seems plausible that if we have a 66 degree F night that lasts 10 hours, we'd have more total activity (proxied by passes) than if the night were only 6 hours. I might be missing part of the question or background knowledge.

11. Re: How to investigate a curvilinear relationship? (power curve?)

That's right; once we account for temperature the effect of night length is negligible, since as we enter winter, nights lengthen but also get colder. Temperature has a strongly reported effect on bat activity since when it's cold bats may not rouse, but also because it impacts insect availability. Even if I wanted to include more on night length I fear prof wouldn't be too happy since he's expressed that he doesn't really want it included in the paper save for a passing comment.

The dependent variable in the scatter plot in my original post is passes per night and that's what I'm looking for help with, but to account for differences in night length when looking at a single night in isolation, the passes are summed by hour (though that's only relevant in the portion of the paper where I discuss change in activity patterns over singular nights).

12. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by crook
That's right; once we account for temperature the effect of night length is negligible, since as we enter winter, nights lengthen but also get colder. Temperature has a strongly reported effect on bat activity since when it's cold bats may not rouse, but also because it impacts insect availability. Even if I wanted to include more on night length I fear prof wouldn't be too happy since he's expressed that he doesn't really want it included in the paper save for a passing comment.
Since the temperature has bioplausibility, I would think to include it regardless, so long as it has support from prior research.

I can understand your PI's desire to leave something out, but is there prior research that would suggest night length isn't important (i.e. can the omission be supported by theory and or research, or would the reviewers just have to trust in your PI's judgement?)? The biggest thing I think would be justification for omitting the night length variable (especially if you're using the raw number of passes).

Could you humor me and just post some output for two things:
1) a regression including both night length and temp
2) regression with just night length

13. Re: How to investigate a curvilinear relationship? (power curve?)

Passes = dead bat? Provide some more general context. Bat dies on cold day, there are fewer bats to die on the next day.

Before you get too involved with nuances, have you examined the model residuals, independence, normality, homoscedasticity. In addition, you can test for multicollinearity between your two IVs. If they have a low Tolerance value, you can drop one since using it will result in very large standard errors around the Y estimate.

Also, I don't see any amazing curve-linear relationship happening in your graph. As I noted, it would be beneficial to plot your fitted line from the model on top of the scatterplot to see how it actually looks.

If you want me to clarify myself, please specify which parts seem Dutch! Once again, define how you collected these values, so we can look for issues, such are do you need to run a Poisson Regression, etc.

14. Re: How to investigate a curvilinear relationship? (power curve?)

Originally Posted by hlsmith
Passes = dead bat? Provide some more general context. Bat dies on cold day, there are fewer bats to die on the next day.

Before you get too involved with nuances, have you examined the model residuals, independence, normality, homoscedasticity. In addition, you can test for multicollinearity between your two IVs. If they have a low Tolerance value, you can drop one since using it will result in very large standard errors around the Y estimate.

Also, I don't see any amazing curve-linear relationship happening in your graph. As I noted, it would be beneficial to plot your fitted line from the model on top of the scatterplot to see how it actually looks.

If you want me to clarify myself, please specify which parts seem Dutch! Once again, define how you collected these values, so we can look for issues, such are do you need to run a Poisson Regression, etc.

This is the original thread. Bat passes refers to flying by a sensor, I believe.

Collinearity doesn't appear to be too terrible (correlation of temperature and night length is moderate, so the VIF would be well below 10, even 5).

Just to clarify, multicollinearity doesn't effect the model standard deviation (standard error of the regression)-- it can inflate the standard errors for the involved coefficients, though.

As you said, the curvilinear pattern doesn't appear very strong at all (what I meant by temp_squared "doing the trick" was that it would get OP to a curvilinear form if that's what he needs).

I think the OP meant he would appreciate if you explained how to get the line of best fit to show on a scatter plot in SPSS.

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