Probability theory and regression analysis are not topics of elementary school. I am afraid that it won't be possible to explain it at such a level without inviting readers to draw wrong conclusions...

What you need is called a (general) linear model that includes a continuous predictor and a categorical predictior (giving the group), and the interaction of these two predictors. The interaction is what you are interested in, it estimates the difference in slopes between the groups. The old-fashioned term for this is ANCOVA: analysis of co-variance. You will find a lot of resources to read in the internet. Unfortunately, the wikipedia article on ANCOVA is not very good.

Here are only few examples:

http://r-eco-evo.blogspot.de/2011/08/comparing-two-regression-slopes-by.html

http://www.theanalysisfactor.com/using-adjusted-means-to-interpret-moderators-in-analysis-of-covariance/

http://www.biostathandbook.com/ancova.html

To really understand what you are doing you would have to learn statistics in general and the theory of linear models in particular. A good practical start could be the book "Data analysis using Regression and Multilevel/Hierarchical Models" be Gelman and Hill (Cambridge).

Probability theory and regression analysis are not topics of elementary school. I am afraid that it won't be possible to explain it at such a level without inviting readers to draw wrong conclusions...

What you need is called a (general) linear model that includes a continuous predictor and a categorical predictior (giving the group), and the interaction of these two predictors. The interaction is what you are interested in, it estimates the difference in slopes between the groups. The old-fashioned term for this is ANCOVA: analysis of co-variance. You will find a lot of resources to read in the internet. Unfortunately, the wikipedia article on ANCOVA is not very good.

Here are only few examples:

http://r-eco-evo.blogspot.de/2011/08/comparing-two-regression-slopes-by.html

http://www.theanalysisfactor.com/using-adjusted-means-to-interpret-moderators-in-analysis-of-covariance/

http://www.biostathandbook.com/ancova.html

To really understand what you are doing you would have to learn statistics in general and the theory of linear models in particular. A good practical start could be the book "Data analysis using Regression and Multilevel/Hierarchical Models" be Gelman and Hill (Cambridge).

In addition to Jochen's comment: interpretation of ANCOVA results may be difficult, since depending on the method you use to test your coefficients, p-values may depend on the order of your variables in the model. Some of them strongly depend on the coding of the variables (but, well, it's already the case in simple regression...).

Moreover, if the range of the continuous predictor is different between the two (or more?) groups, interpretation is also difficult.

Kitartás...

Regression analysis is used to model the relationship between a response variable and one or more predictor variables. one applied of the regression analysis can be used on research in elementary schools

Thanks for all your kind an helpful answers. I think I should clearly state what I want to do: I did a multiple (stepwise) regression for a sample. Until here fine: 2 predictors in the model were significant predicting 39% of the variance in the dependent measure. After that I found group differences in one of the predictors and I have decided to do two regressions (split-data) as based on Any Field and found that in one group the relationship is twice as stronger than in the other group (25% and 50%). Now this is clear until here, but I would like to test whether this difference is statistically significant (if it makes sense?) and whether the slopes are similar (or statistically different?). Many thanks again!

Dear Jochen,

I did the ANCOVA (I understand it a bit :-)), and the interaction between the grouping (fixed) factor and the covariate of interest (that predicted 25% in one and 50% in the other group) was not significant. Does this mean that the slopes do not differ, only the strength of relationship is different in the two groups? If I understand correctly, this may be the interpretation of the ANCOVA interaction?

Dear Attila,

you now describe that you judge the inteaction in your ANCOVA model as "non-significant". This means that you are not surprised (enough) to actually see such differences between the slopes under the assumption that no such dependency between slope and group exists. Read this senstence again if it is unclear, because every word is important here.

Having a non-significant result (for whatever null hypothesis) means nothing more than your data does not provide sufficient "evidence against"  that tested hypothesis. The reasons for this insufficient amount of evidence can not be derived from this data or from this result. In particular, you can't say that the tested hypothesis is wrong or false - that's not possible. All you can say is that you deem your data not sufficiently conclusive to reject the null hypothesis. If you decide that your result is significant, then you say you think the data is too strange under the assumption of the null hypothesis. This means you find the simple model being inadequate to describe the available data sufficiently well, and that it would be sensible to use the more complex model (that is not restricted by the null hypothesis). Again, this is be no means to be taken as a proof of falseness of the null hypothesis ("all model are wrong - but some are useful"!).

So, note that your statement of a "non-significant interaction" is already an interpretation. This is your interpretation saying that you would not use a model including the interaction to describe your data. The result of a significance test is a p-value (the empirical significance of your data under the assumed model). This result needs an interpretation, and saying that values below, say, 0.01 or 0.2 or whatever are considered significant is an interpretation (that may or may not make sense in the particular context).

Many thanks Jochen! Yes, I think we are talking about the same thing. I have no (theoretical) reason to expect interaction between the covariate and the the fixed factors (groups) in the ANCOVA, The p = .323, so it is more than .05. And yes this is  result showing that the association between the IV and DV is stronger in one group than in the other (the covariate is significantly different between the two groups, p

Not sure what you mean with the "strength of the association". The interaction is about the difference in slopes. With p(interaction) = 0.3, the observed difference in slopes is not unexpected under the assumption that there is no interaction (i.e. that the slopes in both groups are the same). Such a large p-value is usually no reason to believe that modelling the interaction is relevant (and to reject the null hypothesis that the difference of slopes is zero).

Noteably, this is NOT indicating that in fact the slopes are exactely equal. It only sais that your data may be described well enough with a model without interaction.

I was simply interested if the slopes are different (good that they are not). What I mean by 'strength of the association' is that the IV explains more of the variance in DV in one group and less in the other group. R sq is .25 in one and .5 in the other.

You are logically inconsistent. For the slopes you claim "good that they are not [different]" becuse the interaction is "not significant", and for the strngth of association (by R²) you claim that there is a difference, but you have not shown that the R² values are statistically different...

But more importantly, the conclusion that the slopes are not different does NOT follow from the non-significant interaction. This result only means that your data is inconclusive regrading the use of an interaction term in a model to describe your data.

You asked a witness if he has seen the accused at the crime scene. The witness says "no, I did not recognise anything strange" (-> large p-value). Now you conclude that the accused is not guilty (-> no interaction). But the problem is that we do not know if the witness had a chance to see the accused (has the witness been at the crime scene, at the time of the crime, was it bright engough, was the accused disguised, was the witness alert and thousand things more to be able to make that connection). If the chance to see and recognize the accused was low, the witness is useless. But this information is not given with the simple statement of the witness (the p-value).

If the witness aid at least something like: "yes, there were strange things going on that I would not have expected when there was no crime", then, and only then, one may conclude that the conditions for the witness to see something must have been good enough (why should he have recognised something otherwise?).

Non-rejection of a null hypothesis is not an indication of the absence of an effect. It only indicates that the available data can be explained sufficiently well with a model restricted to the null hypothesis.

I got the point! Indeed. Can you tell me which result tells me (in SPSS, either regression or ANCOVA) whether the R2  in the two groups are significant (i.e. that .25 is different from .5)? Or should I do some other calculations? Do you want me to send you the Output of the ANCOVA, that perhaps it makes it easier? Many thanks, I really appreciate your attention to this!

Under the assumptions of the ANCOVA model, the test on the interaction (difference of slopes between the groups) is equivalent to the test on the difference of R² between the groups.

Excellent! Many thanks. So, I rephrase, "If my interaction is statistically not significant than the R2 is also not significantly different between the groups." Is that what you mean? The last thing then,  that I do not understand, is that why do I have a group main effect in the covariate,, but no group by covariate interaction? (Again, should I send you the SPSS output for easier interpretation?).

I do not see clearly why you say that having a R² twice higher in a group than in the other is an indication of « higher strength of relation » or any other similar formulation.

The interaction test in an ANCOVA compare the slopes in the two groups, assuming similar distribution [Gaussian] and variance in the two groups. No interaction means the same slope in the two groups, hence same R² is expected. But things are not so simple when looking at the R².

Indeed, this is false. You may well have the same slope in the two groups, but different R² (hence different % of explained variance) just because you have a group with a higher variance (for a fixed X value, of course), or any more complex pattern.

It may also well result from a different range in X values: obviously, the higher the X range, the higher the % of explained values of the Y (assuming a non-null slope in Y = aX + b + epsilon), whatever everything else.

So may be the group with R² = 0.5 simply just has a much larger range of X values than the group with R² = 0.25. Or a smaller variance. Or both. Or a different distribution than a Gaussian. Or who knows what...

I don't see much interest in comparing the R² / % of explained variances per se, it depends on two much things to be easily interpretable.

And to be exact, last Jochen's affirmation of equivalence holds only in the assumption of same variance in the two groups, otherwise a model per group or a common model are not equivalent. A model per group is an easy way to allow different variances per group... (you may compare your two residual variances, by the way, to check this hypothesis).

Let's assume your model is Y = a X + b + epsilon, and both a and b may be different between the two groups.

If you a have a « group » effect in your covariate, that means that the X values have a different range in the two groups (and one is shifted). That is, X is in [0, 1] in one group and in [ 0.5, 2 ] in the other, for instance.

This means that interpretation is difficult, especially if the overlapping is small. It does not mean you should expect different _slopes_ (not even different b, by the way, unlesss you are doing some group-mean centered coding for X).

It also means that it is almost certainly (one of) the reason for different R² in the two groups. Forget R², use your model and it's in-depth interpretation.

Thank you Emmanuel! I will check this too. Indeed, this is what I have, exactly the you phrased it:: "You may well have the same slope in the two groups, but different R² (hence different % of explained variance) just because you have a group with a higher variance (for a fixed X value, of course),"

Köszönöm!:-)

C'est exact:

"If you a have a « group » effect in your covariate, that means that the X values have a different range in the two groups (and one is shifted). That is, X is in [0, 1] in one group and in [ 0.5, 2 ] in the other, for instance." This is the way I interpret my results,... and theoretically this makes sense.

Merci Emmanuel

That's correct: a test on R² is equivalent to a test on the slopes only if the ranges of the x-values for both groups are the same. Thanks Emmanuel!

Nincs mit / You're welcome

I'm now facing the same problem. I want to check whether the R square scores of two groups are significantly or not. Don't know how to do it.

Short answer: do not do that. Think about what your question is in your model terms, and check this instead.

Thank you Emmanuel. I want to examine whether a moderator significantly influence the relationship between an IV and a DV. In this case, what analysis should I do?

You see if the interaction term (between the modifier and the IV) is significant -- i.e., whether the model that includes this interaction term has a significantly better fit than a maximally similar model that does not include that interaction term.

Thank you Stephen. You solved my problem.

Dear Colleagues,

Although the topic is pretty old, I like to recommend you this useful tool which allows for simple calculation of t-test for testing the significance of the difference between two regression slopes.

https://www.danielsoper.com/statcalc/calculator.aspx?id=103

All best,

Jakub

Dear Jakub,

This is very useful!

Thank you so much!

Best regards,

Attila

Hello everyone. I'd like to revive this post a year later with my own problem!

Ill put my question as simply as i can:

I have 2 groups (healthy vs unhealthy), took measures at 7 different walking speeds, and I have 2 parameters (lets call them CV and FP) that i want to examine.

Firstly I want to see how much of the IV (FP) can be predicted by the DV (CV) at each walking speed. My plan was to get the R squared values for each group and each speed separately to see what % of the IV can be explained by the DV. Secondly, i'd like to see how that compares between the two groups, however im not sure how I can make a comparison between the groups this way.

i stumbled upon 'dummy regressions', in that you use a dummy variable to represent the explanatory variable (in this case healthy or unhealthy). would something like that work for me?

I hope this makes sense, any help is greatly appreciated!

First question: are your « 7 different walking speeds » applied to the same patients/animals/individuals or on different ones? That changes a lot the experimental design, hence its statistical analysis!

For the rest: you *always* use what you call "dummy regression" with categorical variables, anyway, so the question on "will that work" is not on that level, but more on issues like the previously mentioned one. And more generally, which variable to enter in the model, checking your model assumptions and so on.

Good luck,

Emmanuel Curis thank you, your answer has already been helpful!

Two separate groups, the same 7 speeds are used for both groups.

I would like to see if a predictive relationship between 1 DV + 1 IV appears throughout the 7 speeds for each group (is it appropriate to say my IV has 7 levels?), and to see how that relationship compares between the two groups (which group shows a stronger relationship).

Thanks again for any reply

The question was not if the 7 speeds were the same for the two groups (which is better for comparisons, but not related to the problem I raised). It was if the 7 speeds were applied to the same individuals.

In other words, was patient 1 of group 1 measured for speed 1, then again for speed 2, then again... ? Or was this patient 1 of group 1 measured for speed 1 only, patient 2 of group 2 for speed 2 only... and so on.

Since speed seems a quantitative variable, and not qualitative, I would not speak of "levels". Instead, I would wonder if the relationship between Y and speed is linear or if another shape is expected or...

After all that, what you suggest as a question really looks like as an interaction analysis, which is quite complex. You should really consult for a local statistician and follow (slightly) advanced curses on the linear model in statistics, to understand what you'll exactly be doing when analyzing your data.

Its the first one you said, participant 1 of group 1 measured for speed 1, then speed 2, then speed 3..... then the same for group 2.

I think i know where to shift my attention to now,

thanks for your time Emmanuel