Anova and interaction with ordinales variables

[Jul Herb]

Hello,

I'm sorry, I'm new to this kind of analysis, but my boss asked me to analyze the interaction between two variables on a third one.

So the two variables are categorical (wealth and work), and we have to see the impact on a third one which is ordinal categorical (never,..., always).

The problem is that my boss really wants us to do a two-way anova.
It seems to me that an anova is usable only if the Y is continuous, which is not the case.
For me, to do this interaction I would have done a regression like that ologit y x1##x2


My boss gave me an example of a type of anova he wants me to do, can you tell me if this is consistent? I have the impression that it's really bad but he insists a lot.

(that's not my true data)

* Example generated by -dataex-. To install: ssc install dataex
clear
input long(work wealth y)
4 2 1
1 3 2
1 3 2
2 3 2
1 3 2
3 1 3
2 1 3
4 1 3
4 1 3
4 1 3
4 1 3
4 1 3
4 1 3
3 1 3
4 1 3
4 1 3
4 2 3
1 2 3
3 2 3
1 2 3
1 2 3
1 2 3
4 2 3
1 2 3
4 2 3
3 3 3
3 3 3
2 3 3
3 3 3
1 3 3
3 3 3
1 3 3
4 3 3
1 3 3
4 3 3
1 3 3
1 3 3
1 3 3
2 3 3
3 1 4
4 1 4
4 1 4
3 1 4
2 1 4
4 1 4
4 1 4
1 1 4
3 1 4
1 1 4
2 1 4
1 1 4
1 1 4
1 1 4
2 1 4
4 1 4
1 1 4
2 1 4
1 1 4
4 1 4
4 1 4
1 1 4
4 1 4
3 1 4
1 1 4
4 1 4
2 2 4
1 2 4
4 2 4
3 2 4
4 2 4
4 2 4
3 2 4
1 2 4
1 2 4
4 2 4
1 2 4
3 2 4
1 2 4
3 2 4
4 2 4
1 2 4
3 2 4
1 2 4
1 2 4
3 2 4
1 3 4
4 3 4
1 3 4
1 3 4
1 3 4
3 3 4
2 3 4
1 3 4
1 3 4
2 3 4
3 3 4
4 1 5
4 1 5
4 1 5
4 1 5
end
label values work activite
label def activite 1 "Engineer", modify
label def activite 2 "Teacher", modify
label def activite 3 "Reseacher", modify
label def activite 4 "Tech", modify
label values wealth village_groupe
label def village_groupe 1 "Cat A", modify
label def village_groupe 2 "Cat B", modify
label def village_groupe 3 "Cat C", modify
label values y peur
label def peur 1 "never", modify
label def peur 2 "rarely", modify
label def peur 3 "sometimes", modify
label def peur 4 "often", modify
label def peur 5 "always", modify

So here's my data, and here's what he wants me to do :

contract y work wealth, zero
sort work wealth y
replace _freq=_freq*y
gen ln_freq=ln(_freq+1)/ln(10)
anova ln_freq work wealth

It's like if he wants to run anova on the logarithm of the number of observations of each category weighted by the variable y (*1 if y==1, *2 if y==2..). He told me he is doing that weight to make y continuous and the log+1 transformation for the homoscedasticity
Is this something that makes sense?


Sorry for this weird question,
Thank you

[Joseph Coveney]

Is this something that makes sense?

Not to me.

You boss's method multiplies the values of the outcome variable by the marginal frequency of the predictor group. So, even if all of the work-activity groups have the identical distribution of fear scores--that is, even if engineers have an average score of 3 and teachers have the identical average score of 3 and researchers have the identical average score of 3 etc.)--then ANOVA will still declare a so-called statistically significant difference if there are many more engineers than teachers and researchers.

However transformed (e.g., logarithm), an outcome for engineers that is Y × 37 (marginal total of engineers in your example dataset) will be seen by ANOVA as different from one for teachers that is Y × 11 (marginal total of teachers in your dataset), despite both work-activity groups' having the identical (Y) outcome score on average.

I think that you can show this phenomenon to your boss by performing a simulation under the null hypothesis (identical average y, say, distributed uniformly from 1 to 5, using replace y = runiformint(1, 5) in each replication) with the observed marginal frequencies for work-activities and village-group categories.

[Joseph Coveney]

Actually, your boss is multiplying by the conditional (cell) frequencies and not the marginal frequencies, but I think that the same phenomenon would obtain: even if each cell's average outcome value is identical, but the cell counts vary, then the outcome variable will be distorted.

[Jul Herb]

Thanks a lot.

Any ideas to show the interaction between work and wealth on Y ? I never worked with orderer y, an ologit is usable?

[Joseph Coveney]

Depending upon your sample size, you could go with what you show above with ologit. There are some assumptions, for example, proportional odds, but I wouldn't sweat that too much.

With smallish sample sizes, you could consider a permutation test that is based upon the corresponding factorial ANOVA.

I illustrate both below on your fake dataset. With your sample size (truncated by the default setting of dataex to 100 observations), both approaches give the same answer, viz., P = 0.8 for the interaction term.

.ÿ
.ÿversionÿ17.0

.ÿ
.ÿclearÿ*

.ÿ
.ÿ//ÿseedem
.ÿsetÿseedÿ1605085412

.ÿ
.ÿquietlyÿinputÿbyte(workÿwealthÿy)

.ÿ
.ÿologitÿyÿi.work##i.wealth,ÿnolog

OrderedÿlogisticÿregressionÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿÿÿÿ100
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿLRÿchi2(11)ÿÿÿ=ÿÿ16.95
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿ=ÿ0.1095
Logÿlikelihoodÿ=ÿ-90.60301ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿPseudoÿR2ÿÿÿÿÿ=ÿ0.0855

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿyÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿworkÿ|
ÿÿÿÿÿÿÿÿÿÿ2ÿÿ|ÿÿ-.8556323ÿÿÿ1.205031ÿÿÿÿ-0.71ÿÿÿ0.478ÿÿÿÿ-3.217449ÿÿÿÿ1.506185
ÿÿÿÿÿÿÿÿÿÿ3ÿÿ|ÿÿ-1.369716ÿÿÿ1.113499ÿÿÿÿ-1.23ÿÿÿ0.219ÿÿÿÿ-3.552133ÿÿÿÿ.8127014
ÿÿÿÿÿÿÿÿÿÿ4ÿÿ|ÿÿ-.9753956ÿÿÿ.9072945ÿÿÿÿ-1.08ÿÿÿ0.282ÿÿÿÿÿ-2.75366ÿÿÿÿÿ.802869
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿwealthÿ|
ÿÿÿÿÿÿÿÿÿÿ2ÿÿ|ÿÿ-1.551978ÿÿÿ.9410824ÿÿÿÿ-1.65ÿÿÿ0.099ÿÿÿÿ-3.396466ÿÿÿÿ.2925095
ÿÿÿÿÿÿÿÿÿÿ3ÿÿ|ÿÿ-2.695067ÿÿÿ.9470412ÿÿÿÿ-2.85ÿÿÿ0.004ÿÿÿÿ-4.551234ÿÿÿ-.8389002
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿwork#wealthÿ|
ÿÿÿÿÿÿÿÿ2ÿ2ÿÿ|ÿÿÿÿ2.40761ÿÿÿ2.590166ÿÿÿÿÿ0.93ÿÿÿ0.353ÿÿÿÿ-2.669022ÿÿÿÿ7.484243
ÿÿÿÿÿÿÿÿ2ÿ3ÿÿ|ÿÿÿ.8556323ÿÿÿ1.591631ÿÿÿÿÿ0.54ÿÿÿ0.591ÿÿÿÿ-2.263908ÿÿÿÿ3.975173
ÿÿÿÿÿÿÿÿ3ÿ2ÿÿ|ÿÿÿ2.305202ÿÿÿ1.502107ÿÿÿÿÿ1.53ÿÿÿ0.125ÿÿÿÿ-.6388737ÿÿÿÿ5.249278
ÿÿÿÿÿÿÿÿ3ÿ3ÿÿ|ÿÿÿ1.605752ÿÿÿ1.439831ÿÿÿÿÿ1.12ÿÿÿ0.265ÿÿÿÿ-1.216265ÿÿÿÿÿ4.42777
ÿÿÿÿÿÿÿÿ4ÿ2ÿÿ|ÿÿÿ.5601647ÿÿÿ1.243726ÿÿÿÿÿ0.45ÿÿÿ0.652ÿÿÿÿ-1.877493ÿÿÿÿ2.997823
ÿÿÿÿÿÿÿÿ4ÿ3ÿÿ|ÿÿÿ1.211432ÿÿÿ1.489148ÿÿÿÿÿ0.81ÿÿÿ0.416ÿÿÿÿ-1.707244ÿÿÿÿ4.130109
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿ/cut1ÿ|ÿÿ-6.452018ÿÿÿ1.281771ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-8.964244ÿÿÿ-3.939793
ÿÿÿÿÿÿÿ/cut2ÿ|ÿÿ-4.755205ÿÿÿÿ.914072ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-6.546753ÿÿÿ-2.963657
ÿÿÿÿÿÿÿ/cut3ÿ|ÿÿ-2.013954ÿÿÿ.7856569ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-3.553813ÿÿÿ-.4740944
ÿÿÿÿÿÿÿ/cut4ÿ|ÿÿÿ2.013954ÿÿÿ.7856569ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4740944ÿÿÿÿ3.553813
------------------------------------------------------------------------------

.ÿtestparmÿi.work#i.wealth

ÿ(ÿ1)ÿÿ[y]2.work#2.wealthÿ=ÿ0
ÿ(ÿ2)ÿÿ[y]2.work#3.wealthÿ=ÿ0
ÿ(ÿ3)ÿÿ[y]3.work#2.wealthÿ=ÿ0
ÿ(ÿ4)ÿÿ[y]3.work#3.wealthÿ=ÿ0
ÿ(ÿ5)ÿÿ[y]4.work#2.wealthÿ=ÿ0
ÿ(ÿ6)ÿÿ[y]4.work#3.wealthÿ=ÿ0

ÿÿÿÿÿÿÿÿÿÿÿchi2(ÿÿ6)ÿ=ÿÿÿÿ3.19
ÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿ=ÿÿÿÿ0.7843

.ÿ
.ÿpermuteÿyÿFÿ=ÿe(F_3),ÿreps(3000)ÿnodots:ÿanovaÿyÿwork##wealth

MonteÿCarloÿpermutationÿresultsÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobservationsÿ=ÿÿÿ100
Permutationÿvariable:ÿyÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿpermutationsÿ=ÿ3,000

ÿÿÿÿÿÿCommand:ÿanovaÿyÿwork##wealth
ÿÿÿÿÿÿÿÿÿÿÿÿF:ÿe(F_3)

-------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿMonteÿCarloÿerror
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-------------------
ÿÿÿÿÿÿÿÿÿÿÿTÿ|ÿÿÿÿT(obs)ÿÿÿÿÿÿÿTestÿÿÿÿÿÿÿcÿÿÿÿÿÿÿnÿÿÿÿÿÿpÿÿSE(p)ÿÿÿ[95%ÿCI(p)]
-------------+-----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿFÿ|ÿÿ.5479445ÿÿÿÿÿÿlowerÿÿÿÿÿ736ÿÿÿÿ3000ÿÿ.2453ÿÿ.0079ÿÿ.2300ÿÿ.2611
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿupperÿÿÿÿ2264ÿÿÿÿ3000ÿÿ.7547ÿÿ.0079ÿÿ.7389ÿÿ.7700
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿtwo-sidedÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.4907ÿÿ.0091ÿÿ.4728ÿÿ.5086
-------------------------------------------------------------------------------
Notes:ÿForÿlowerÿone-sidedÿtest,ÿcÿ=ÿ#{Tÿ<=ÿT(obs)}ÿandÿpÿ=ÿp_lowerÿ=ÿc/n.
ÿÿÿÿÿÿÿForÿupperÿone-sidedÿtest,ÿcÿ=ÿ#{Tÿ>=ÿT(obs)}ÿandÿpÿ=ÿp_upperÿ=ÿc/n.
ÿÿÿÿÿÿÿForÿtwo-sidedÿtest,ÿpÿ=ÿ2*min(p_lower,ÿp_upper);ÿSEÿandÿCIÿapproximate.

.ÿ
.ÿexit

endÿofÿdo-file


.

[Jul Herb]

Thank you very much, this will help me a lot.
I'm trying the first one (ologit), because i have more than 500 people in my database (i didn't know dataex was truncated at 100 observations).