Hay que ver el R!!

En el statistica es un botoncito mas del programa al hacer el test.

Basicamente, solo puedes hacer el ancova bien, si las pendientes de los dos grupos  son similares. Por ej,para poder ver si los machos son mas pesados de lo esperado que las hembras para un tamaño similar (que es lo que te testa el ancova, diferencia de intercepts)  comparas si la relacion peso-tamaño varia entre sexos, pero para que el test sea fiable los machos y hembras deben aumentar de peso en relacion a su talla de forma similar (slopes similares),

Bests

Jose

• My question is, can the interaction term be interpreted as a test for the homogeneity of regression slopes?

Yes.

The interaction estimates the difference in the slopes. If you type

summary(ancova)

you will get (among other things) a printout of the coefficients of your model and their standard errors. With

confint(ancova)

you will get the confidence intervals for the coefficients. You can use this information to see how large the estimated difference in the slopes is and if it may be relevant.

Heath Stearns, in R the shorthand for entering each main effect and interaction is x1 * x2. So, Mario has entered what you're suggesting. To do it your way you would typically change the "*" to a ":". But R figures out you're being redundant and does the right thing (usually).

No, the interaction term is not a test for homogeneity, it is a test for the lack of homogeneity. The slopes could be very heterogenous and you fail to detect the distinction simply due to a low N. You can't demonstrate the homogeneity of the slopes this way. The only way to do that is descriptively and with theory. Plot your data with size predicting weight and look at the slopes of each sex separately. Make a rational judgment of similarity. The predict function in R will allow you to get separate predicted values for each sex to plot.

Ok, to resolve the contradiction in Johns and my answer:

The hypothesis underlying the test of the interaction is that the difference in slopes is zero, what I would understand as "homogeneity of slopes". This is the tested hypothesis. However, the p-value provided by a test is not an indicator of the homogeneity. It may only be used as an indicator of heterogeneity, because large p-values only say that similarily "extreme" test statistics are not unexpected under the assumption (-> equal slopes), but it does by no means exclude any other reasons for this result (e.g. the variance is too large to see anything relevant, or the sample size is too small to see anything relevant, etc.). Things become interesting when the p-values is really small. This means that either the data is "strange", representing a rare ("unlucky") sample, or it is not well explained by the hypothesis (equal slopes). So if the best is done to obtain "non-strange", typical, representative data and the rest of the model and assumptions are sensible, then this points to a problem with the hypothesis.

At the end, the question IF the slopes are equal or not is a philosophical question that will not be answered with any empirical data. The more relevant and scientific question is: HOW much different are the slopes? What is the size of the estimate, and how precise is this estimate? Is the precision high enough to discern a relvant from an irrelevant difference?

I want to add that saying "you can not use ANCOVA without homogenity of slopes" is not necessarily correct. I like more think about it in a mechanistic way. You want to use ANCOVA to compare the response of a dependent variable in two groups (e.g. treatment vs. control) ) which co-varies with another continous variable (e.g. body weight). A classic example in fish is when you look at liver weight in response to chemical exposure, because liver weight itself is known to covary with body weight (i.e. bigger fish -> bigger liver obviously). You need to have homogenous slopes to make sure you can detect a difference between the two groups regardless of the co-variable. If homogenity of slopes is not given (and I agree with Jochen that the important question is how different the slopes are, not just the p) it could also simply mean that the effect of a treatment  differs for individuals depending on e.g. body weight. So, yes staistically the interaction term can be used as a measure of homogenity of slopes, but I strongly recommend to ask yourself the question what do your results mean in your context and not just to rely on p-values to conclude from your data.

Christian Michell, your example with the fish sounds more like correlated predictors than heterogenous slopes. 

You're correct if one just interprets ANCOVA as multiple regression but technically it's incorrect to perform ANCOVA with heterogeneity of slopes, or correlated predictors because the purpose of ANCOVA as a technique is to be able to have a simple reduction in the variability of the response variable and none of those more complicated regression issues. In practice an ANCOVA is almost never what you should be doing but, as Christian implies, you should be doing multiple regressions where you carefully examine the "covariate" (which is really just another predictor).

Once you no longer has the (over)simplification of ANCOVA interpretation with homogeneous slopes and independent covariate you shouldn't be calling it ANCOVA anymore but that doesn't preclude reporting the full multiple regression.

Thanks a lot to everybody. Of course I am not going to support completely in a P-value. My question was more to know the significance of this interaction term as I saw that in some programs are included but not in others. Your answers have been really helpful. Thanks again

So, another issue. Anybody knows if the STATISTICA test and the interaction term in R compute the same statistics?

Thanks

They're both tests of interaction. With R you set it up yourself so in order to make it match you need to look at the Statistica manual.

http://documentation.statsoft.com/STATISTICAHelp.aspx?path=Gxx/Grm/Dialogs/VisualGSRQuickTabHomogeneityofSlopesModel

Dear Nina,

the coefficient estimate (and its CI) of the interaction (Pdose:Pefficiency) tells you just that.

If Pefficiency has two levels, then there should be one coefficient for the interaction, and this is the difference between the slopes (TBARS vs. Pdose), as you will have two lines (one per level of Peff). The coefficient for Pdose you reported is the slope for the reference level of Peff (what is coded as 0, usually). The slope for the other level of Peff is the sum of that slope and the interaction.

The ANCOVA model is:

Y = b0 + b1*Xeff + b2*Xdose + b3*Xeff*Xdose

As Pdose is continuous, Xdose = Pdose (no transformation applied), and Xeff is the 0/1 coded design variable for Peff (Xeff=0 when Peff is the reference level and Xeff=1 when Peff is the other level).

b2 is the slope for the reference level of Peff (when Xeff=0).

b3 is the difference in slopes between the levels of Peff.

b2+b3 is the slope for the other level of Peff (when Xeff=1).

and

b0 is the intercept for the reference level of Peff.

b1 is the difference in the intercept between the levels of Peff.

b0+b1 is the intercept for the other level of Peff.

So what you are looking for is the estimate for b3. STATISTICA should give you this value along with its SE and CI and a p-value (testing b3=0). I am not sure if the "homogeneity of slopes"-analysis will provide this, but I know that you can simply fit GRMs and you will get all that information. You need a table of the regression coefficients. In the Statistica help I found this sentence:

Regression coefficients. Click the Regression coefficients button to display a spreadsheet of the current parameter estimates (b), standardized parameter estimates (Beta), their standard errors, significance levels, and related statistics. [...] Please see the Coefficients option on the GRM Results - Quick tab for detailed information.

Does this answer your question?

1) Not sure why you distinguish GRM (general regression model) and GLM (general linear model). To me they are synonyms. The order of factors in a model matters only if you do series of F-tests and when the design in unbalanced. Shouldn't matter in your case.

2) The fitted coefficients mean:

• the intercept of the line for the Peff=0 group is 29.9, and this line has a slope of -23.8.
• the intercept of the line for the Peff=1 group is 3.1 units higher than that of the Peff=0 group (the intercept is thus 29.9+3.1 = 33.0).
• the slope of the line for the Peff=1 group is -13.1 lower than that of the Peff=0 group (the slope is thus -23.8-13.1 = -36.9)

I attached a plot showing this graphically.

3) You would still add "Pdose" and "Peff" (now categorical with 3 levels) to the model, including their interaction. The model will then fit 3 lines (-> two coefs. for intercept-differences and two coefs for the interaction [slope differences]). I don't know how to do that in Statistica; I use R ;)