Interaction Between Continuos Variable and Binary Variable

[alessio carozzino]

Hi everybody,

I am Alessio Carozzino and I am finishing my Master Thesis. I have just built a model with an Interaction between a categorical variable and a continuous variable. My Categorical variable goes from 1 to 5, while the continuos variable is a CSR Performance Indicator. I hava some problem with the interpretation of results. Below i post the results:

regress MediaFavorability i.CSSPOrientation##c.ESGt2 TotNews ROA Leverage Size Employees TobinsQ TotalDonations i.LocalvsGlobal i.Year, vce (cluster com)

CSSPOrientation
2	2.347701	.7718055	3.04	0.003	.8184673	3.876935
3	2.174509	.8457244	2.57	0.011	.4988146	3.850204
4	2.17052	.6512696	3.33	0.001	.8801131	3.460927
5	1.721284	.7230449	2.38	0.019	.2886635	3.153905
ESGt2	.017118	.0090764	1.89	0.062	-.0008658	.0351018
CSSPOrientation#c.ESGt2
2	-.0309832	.0126467	-2.45	0.016	-.0560411	-.0059253
3	-.0285868	.0113146	-2.53	0.013	-.0510052	-.0061685
4	-.0233678	.0106282	-2.20	0.030	-.0444262	-.0023093
5	-.01858	.0111973	-1.66	0.100	-.040766	.003606
TotNews	-.1972081	.1106794	-1.78	0.077	-.4165052	.022089
ROA	2.830913	1.16048	2.44	0.016	.5315711	5.130255
Leverage	.0732461	.5661101	0.13	0.897	-1.048428	1.194921
Size	-.1475866	.0814661	-1.81	0.073	-.3090012	.0138281
Employees	.1008149	.076579	1.32	0.191	-.0509165	.2525464
TobinsQ	-.0107199	.0067058	-1.60	0.113	-.0240066	.0025668
TotalDonations	.0177869	.0111231	1.60	0.113	-.0042521	.0398259
1.LocalvsGlobal	.4081854	.2291409	1.78	0.078	-.045828	.8621988

In my interpretation, CSSP Orientation represents a dummy of each variable against "1" which is the omitted variable. In relation to the slope, I have interpreted it in this way: coefiicient represents the different impact of each dummy when ESG score increase by one unit.

Is it correct?

Thank you very much for your help,

Alessio

[Clyde Schechter]

CSSP Orientation represents a dummy of each variable against "1" which is the omitted variable.

Well, yes, but only conditional on ESGt2 == 0. Because of the interaction in the model, the differences among the CSSPOrientation categories depends on the value of ESGt2, and the coefficients you see are the values that correspond to ESGt2 = 0.

In relation to the slope, I have interpreted it in this way: coefiicient represents the different impact of each dummy when ESG score increase by one unit.

I think you are saying this about the coefficients of the interaction term. The language you are using is imprecise and I cannot be certain what you mean by it, so I can neither confirm nor dispute it.

Let's just go over the interpretation of this model. For simplicity I will ignore all of the variables other than CSSPOrientation and ESGt2. The regression equation says directly:

Media Favorability = constant term + bCSSPOrientation category + bESGT2*ESGt2 + cCSSPOrientation categoryESGt2 + other terms + error

which can be rewritten as two separate equations:

Media Favorability = constant term + bCSSPOrientation category + BESGt2 * ESGt2 + other terms + error, where
BESGt2 = bESGt2 + cCSSPOrientation category

So the final equation says that the slope of the Media Favorability: ESGt2 relationship is the sum of two components: b_ESGt2, which is the same regardless of the CSSP Orientation cateogry, and c_{CSSPOrientation category}, which is different for the different CSSP Orientation categories. This combined slope, B_ESGt2, represents the expected difference in Media Favorability associated with a unit difference in ESGt2 for entities in the particular category of CSSP Orientation. You can, if you like, calculate these sums using the -lincom- command. But there is an easier way to interpret models like this using the -margins- command.

First you need to pick a range of values of ESGt2 that is of interest. For the purposes of illustration, I'll assume that ESGt2 runs from -50 to +50 and that the entire range is of interest. So I would pick several values that span that range.

margins CSSPOrientation, at(ESGt2 = (-50(10)50))
marginsplot

will show you the expected value of Media Favorability in each CSSP Orientation at the values of ESGt2 ranging from -50 to 50 in steps of 10. The -marginsplot- command, which, by the way, must be run immediately after -margins-, will give you a graphical view of this. The plot will show you five line graphs, one for each CSSP Orientation category (including the omitted reference category). They will run at different slopes, and they may or may not intersect within the range of the graph.

In terms of the marginal effects (slopes), you can get those with:

margins CSSPOrientation, dydx(ESGt2)

The output you get using -margins- in this way is, I think, easier to understand and work with, than the direct regression output. Also, it is easy to make mistakes writing out all the different -lincom- commands to get the slopes, whereas -margins- is pretty foolproof.

[alessio carozzino]

Dear Clyde,

thank you for your suggestion. In relation to

In relation to the slope, I have interpreted it in this way: coefiicient represents the different impact of each dummy when ESG score increase by one unit.

, i said this about interaction term.
I suggested that, an increase of 1 unit of ESGt2 generated an additional positive impact for CSSP Orientation1 (that is omitted) in relation to the others partnerships. In fact, the coefficients of each Interaction Terms are negative.
Could it be a possible interpretation?

Alessio

[alessio carozzino]

Dear Clyde,

I have just run

margins CSSPOrientation, dydx(ESGt2)

and I obtained this output:

	Delta-method
dy/dx	Std. Err.	t	P>t	[95% Conf.	Interval]
ESGt2
CSSPOrientation
1	.017118	.0090764	1.89	0.062	-.0008658	.0351018
2	-.0138653	.0090579	-1.53	0.129	-.0318123	.0040818
3	-.0114689	.0113526	-1.01	0.315	-.0339626	.0110249
4	-.0062498	.0058974	-1.06	0.292	-.0179348	.0054352
5	-.001462	.0076269	-0.19	0.848	-.0165738	.0136498

Can I suggest that CSSP Orientation 1 is the only Partnership that benefits, in term of Media Favorability, of an increasement of the ESG Score? The coefficient is positive and it is significant at 10%.

Thank you very much, it is very important for my Master Thesis

Alessio

[Clyde Schechter]

Well, not knowing what your variables mean, I cannot judge what is a "benefit" and what might be a "harm." From a purely statistical perspective, without value judgments, it would be appropriate to interpret these results as follows:

In CSSP Orientation category 1, increasing values of ESGt2 are associated with increasing values of Media Favorability. In the other categories of CSSP Orientation, the estimated Media Favorability:ESGt2 relationships are negative, but the wide confidence intervals imply that these effects are imprecisely estimated, so that the data do not enable us to confidently (at the 10% level) draw conclusions about the direction of the actual association.

[alessio carozzino]

Dear Clyde,
thank you very much for your input.

I would like to ask you another things. With

margins CSSPOrientation, dydx(ESGt2)

, I have demonstrated that In CSSP Orientation category 1, increasing values of ESGt2 are associated with increasing values of Media Favorability, while it is not possibile saying the same for other categories.
However, if I run the following code

margins CSSPOrientation, at(ESGt2 = (22(10)95))

, I obtain this results:

	Delta-method
Margin	Std. Err.	t	P>t	[95% Conf.	Interval]
_at#CSSPOrientation
1 1	1.07704	.3514785	3.06	0.003	.3806302	1.773449
1 2	2.742101	.3863588	7.10	0.000	1.976581	3.507622
1 3	2.621708	.6505686	4.03	0.000	1.33269	3.910726
1 4	2.732708	.3216744	8.50	0.000	2.095352	3.370065
1 5	2.388959	.4328479	5.52	0.000	1.531327	3.246592
2 1	1.24822	.2743211	4.55	0.000	.7046874	1.791752
2 2	2.603449	.2998398	8.68	0.000	2.009355	3.197543
2 3	2.50702	.5399552	4.64	0.000	1.437168	3.576872
2 4	2.67021	.2681412	9.96	0.000	2.138923	3.201498
2 5	2.374339	.3615775	6.57	0.000	1.657919	3.090758
3 1	1.419399	.2084306	6.81	0.000	1.006421	1.832378
3 2	2.464796	.2166651	11.38	0.000	2.035502	2.89409
3 3	2.392331	.4308595	5.55	0.000	1.538638	3.246024
3 4	2.607712	.2174419	11.99	0.000	2.176879	3.038545
3 5	2.359719	.2928367	8.06	0.000	1.7795	2.939937
4 1	1.590579	.1676631	9.49	0.000	1.258376	1.922782
4 2	2.326144	.142803	16.29	0.000	2.043198	2.609089
4 3	2.277643	.3248147	7.01	0.000	1.634064	2.921221
4 4	2.545214	.1720993	14.79	0.000	2.204221	2.886207
4 5	2.345098	.2289157	10.24	0.000	1.891531	2.798665
5 1	1.761759	.1710405	10.30	0.000	1.422864	2.100654
5 2	2.187491	.1012457	21.61	0.000	1.986886	2.388097
5 3	2.162954	.2261538	9.56	0.000	1.714859	2.611049
5 4	2.482716	.1375187	18.05	0.000	2.21024	2.755192
5 5	2.330478	.175173	13.30	0.000	1.983395	2.677561
6 1	1.932939	.2165066	8.93	0.000	1.503959	2.361919
6 2	2.048839	.1285216	15.94	0.000	1.79419	2.303488
6 3	2.048265	.1502091	13.64	0.000	1.750645	2.345886
6 4	2.420218	.1231282	19.66	0.000	2.176255	2.664181
6 5	2.315858	.1435367	16.13	0.000	2.031458	2.600257
7 1	2.104118	.2845553	7.39	0.000	1.540309	2.667928
7 2	1.910186	.1979747	9.65	0.000	1.517925	2.302447
7 3	1.933577	.1405572	13.76	0.000	1.655081	2.212073
7 4	2.35772	.1355202	17.40	0.000	2.089204	2.626236
7 5	2.301238	.1488425	15.46	0.000	2.006325	2.59615
8 1	2.275298	.3626908	6.27	0.000	1.556673	2.993924
8 2	1.771534	.2797842	6.33	0.000	1.217177	2.32589
8 3	1.818888	.2067031	8.80	0.000	1.409333	2.228444
8 4	2.295222	.1688989	13.59	0.000	1.96057	2.629874
8 5	2.286617	.1879879	12.16	0.000	1.914143	2.659091

Results confirm that CSSP Orientation 1 benefit in Term of Media Favorability from increasing ESG Score. At Low ESG Score, they have the lowest margin, while at High Level of ESG Score it reduces this gap. My question is: Can I use this results after that

margins CSSPOrientation, dydx(ESGt2)

showed that is not possible define a causal relationship for other categories?

I hope it is clear!!

Thank you very much,

AC

[Weiwen Ng]

Dear Clyde,
thank you very much for your input.

I would like to ask you another things. With , I have demonstrated that In CSSP Orientation category 1, increasing values of ESGt2 are associated with increasing values of Media Favorability, while it is not possibile saying the same for other categories.
However, if I run the following code, I obtain this results: ...Results confirm that CSSP Orientation 1 benefit in Term of Media Favorability from increasing ESG Score. At Low ESG Score, they have the lowest margin, while at High Level of ESG Score it reduces this gap. My question is: Can I use this results after that showed that is not possible define a causal relationship for other categories?

I hope it is clear!!

Thank you very much,

AC

Alessio, just a note: the output is hard to read. You can actually put the output in the code delimiters like you did for your commands, and it will be more readable.

In addition, a long table with a lot of text is still not that clear, and you can type

marginsplot

to show the result visually. Chances are this will be a lot easier to read. Note, if you want to post it to this forum, please save the resulting graph in PNG format, and post it as an attachment. Per the FAQ, we ask people not to attach .gph files, because readers have to open Stata to access them, whereas an attached PNG file will be visible in the post.

[Clyde Schechter]

Can I use this results after that
margins CSSPOrientation, dydx(ESGt2)
showed that is not possible define a causal relationship for other categories?

That depends on what your original research question was, and how you plan to use these results.

First, I strongly agree with Weiwen Ng's point that these results are much easier to understand as a graph than in a long table. If you plan to use these results, I would display them graphically.

Next, you are now introducing the question of causality into the discussion. Causal inference can never be based purely on data from a single observational study; if you wish to draw causal conclusions, or, for that matter, assert non-causation, you must do so based on a causal model that derives from information external to this analysis.

If your original research goal was to look specifically at category 1, then it is appropriate to report just results from category 1. If your original goal was to look at CSSP Orientation as a whole, then the scientifically appropriate way to present your findings is to show the results for all of the categories.

Finally, you should not confuse statistical significance with causality, nor even with association. Statistical significance is a complicated phenomenon that results first from imposing an arbitrary (and usually indefensible) cutoff on a test statistic that is itself a mixture of information about sample size, variability of the outcome, reliability with which it is measured, and the strength of the associations of the predictor variables in the model with the outcome. So, statistical significance is very far removed from the associations of the predictors with the outcome and is confounded by many things that are not really of interest but which are methodological issues in your study design. In particular, it makes no sense to say that you have an association for category 1 but not for categories 2 and 3 when the effects you are finding for the latter two categories are approximately the same as what you found for category 1. The difference in statistical significance among these effects probably arises from differences in the number of observations in the different categories, or perhaps differential measurement reliability across categories. Also, if you are using the 0.10 significance level, you must also bear in mind that a p-value of 0.129 is only slightly above that threshold. Unless you have a compelling reason to believe that 0.10 significance is exactly the "right" significance level for your study, it is best not to apply these cutoffs so literally. And if you do really believe the 0.10 significance level is really justified, and not just an arbitrary round number that is convenient to use, then you should follow through with that belief by using 90% confidence limits rather than 95% confidence limits throughout the analysis. See -help set level-. And in that case, I would focus my attention on the effect estimates themselves and their 90% confidence intervals.