Three Way Interaction Analysis with All Continous Variables and Testing for Slope Differences

[Ferdi Djajadisastra]

My question in this post refers to this particular website: https://stats.oarc.ucla.edu/stata/fa...tion-stata-12/

It is important to note that I am still a junior researcher with limited knowledge of really advanced statistical methods.

I am doing a research project using the three-way interaction statistical analysis approach (moderating regression analysis). I am using this particular UCLA website to guide my analysis.

My question is regarding the Table below this portion:
/* compute pairwise differences in simple slopes */
What is the meaning of dy/dx in the Table? Is it a Cohen's d score? I asked this question because the editor requested me to: "testing for slope differences". I wondered if this is the statistical analysis the editor referred to for slope differences. Thank you for everyone's attention!

[George Ford]

It is estimating the slope at different values of the interacted terms.

If you have,

Y = b1*X *W *Z

Then dY/dX is b1*W*Z

So you compute at different values of W and Z.

[Ferdi Djajadisastra]

It is estimating the slope at different values of the interacted terms.

If you have,

Y = b1*X *W *Z

Then dY/dX is b1*W*Z

So you compute at different values of W and Z.

Thanks for the response, George. I think my question is actually what should I write for the headings of the Table for the dY/dX coefficient? Can I write it as Cohen's d? Thank you!

[George Ford]

It's not Cohen's D, so I wouldn't. Cohen's D is a standardized difference.

What you have is a slope at different values of the interacted variables.

"Marginal Effect of X at Z"

In some fields, they are called "Simple Slopes".

[Bruce Weaver]

The dydx() function tells -margins- to report the derivative of y with respect to x--in other words the slope. From that UCLA page:

We will use the margins comand to compute the simple slopes for the four values of w and z. We use the dydx option to obtain the slopes. Please note, we have manually annotated the output with the various combinations of w and z in parentheses.

Notice too that an earlier -margins- table shows the 4 simple slopes. You can use the values in that table to compute "by hand" the same pairwise comparisons you get from the -margins- command you asked about. E.g.,

. /*
> margins, dydx(x) at(w=($Hw $Lw) z=($Hz $Lz)) vsquish
>
> Average marginal effects                          Number of obs   =        200
> Model VCE    : OLS
>
> Expression   : Linear prediction, predict()
> dy/dx w.r.t. : x
> 1._at        : z               =    61.75089
>                w               =    62.01345
> 2._at        : z               =    61.75089
>                w               =    43.27655
> 3._at        : z               =    41.94911
>                w               =    62.01345
> 4._at        : z               =    41.94911
>                w               =    43.27655
>
> ------------------------------------------------------------------------------
>              |            Delta-method
>              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
> -------------+----------------------------------------------------------------
> x            |
>          _at |
>           1  |   .0356648   .0951734     0.37   0.708    -.1508717    .2222012
>           2  |   .1435565   .1371306     1.05   0.295    -.1252146    .4123276
>           3  |    .498484   .1884961     2.64   0.008     .1290384    .8679296
>           4  |   .0929561   .1236172     0.75   0.452    -.1493291    .3352413
> ------------------------------------------------------------------------------
> THIS TABLE SHOWS the 4 SIMPLE SLOPES */
.
. /* compute pairwise differences in simple slopes
>
> margins, dydx(x) at(w=($Hw $Lw) z=($Hz $Lz)) vsquish pwcompare(effects)
>
> Pairwise comparisons of average marginal effects
> Model VCE    : OLS
>
> Expression   : Linear prediction, predict()
> dy/dx w.r.t. : x
> 1._at        : z               =    61.75089
>                w               =    62.01345
> 2._at        : z               =    61.75089
>                w               =    43.27655
> 3._at        : z               =    41.94911
>                w               =    62.01345
> 4._at        : z               =    41.94911
>                w               =    43.27655
>
> ------------------------------------------------------------------------------
>              |   Contrast Delta-method    Unadjusted           Unadjusted
>              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
> -------------+----------------------------------------------------------------
> x            |
>          _at |
>      2 vs 1  |   .1078917   .1559956     0.69   0.489     -.197854    .4136375
>      3 vs 1  |   .4628192   .1955057     2.37   0.018     .0796352    .8460033
>      4 vs 1  |   .0572913   .1557822     0.37   0.713    -.2480361    .3626188
>      3 vs 2  |   .3549275   .2449055     1.45   0.147    -.1250785    .8349335
>      4 vs 2  |  -.0506004   .1635762    -0.31   0.757    -.3712038    .2700031
>      4 vs 3  |  -.4055279   .2096843    -1.93   0.053    -.8165015    .0054457
> ------------------------------------------------------------------------------
> THIS TABLE SHOWS PAIRWISE COMPARISONS AMONG THE 4 SIMPLE SLOPES
> */
.
. * COMPUTE THOSE PAIRWISE COMPRISONS "BY HAND" USING SLOPES FROM THE 1ST TABLE
. display _newline ///
> "2 vs 1: ".1435565 - .0356648  _newline ///
> "3 vs 1: ".498484 - .0356648  _newline ///
> "4 vs 1: ".0929561 - .0356648  _newline ///
> "3 vs 2: ".498484 - .1435565  _newline ///
> "4 vs 2: ".0929561 - .1435565  _newline ///
> "4 vs 3: ".0929561 - .498484

2 vs 1: .1078917
3 vs 1: .4628192
4 vs 1: .0572913
3 vs 2: .3549275
4 vs 2: -.0506004
4 vs 3: -.4055279

I hope this helps to clarify things.

[Ferdi Djajadisastra]

Thank you Bruce and George for the clarification! Woops, I guess I made an embarassing mistake in my manuscript. So, I need to change "Cohen's d" in the Table heading with "dy/dx coefficient". Once again, thank you very much for the clarification. It is sometimes difficult to navigate the latest development of statistical analysis.

[George Ford]

Cohen's D is a like a t-statistic unadjusted for sample size, also called a standardized difference.