ANCOVA: Interpret the JMP ANCOVA result. (4 possible combination)

[#1]

Hello all.

I am having problem with interpretating the ANCOVA in JMP.

Problem 1.

Can I use IV as "continuous variable" & "DV as continuous, CV as categorical" ?

Since I had read many reference that saying "ANCOVA need IV-categorical, DV-continuous, CV-continuous".

It is very confusing, since if IV is categorical, IV and DV relationship can not be defined as linear regression... they are just ANOVA...

isn't it??

Problem 2.

There are 4 possible JMP output in ANCOVA

1- slope different, intercept different

2- slope same, intercept same

3- slope different, intercept same

4- slope same, intercept different

How can I interpret 1 and 3 cases?

There seems to be the other approach when this situation comes in,

but it is too complicated.

Please help me.

[#2]

Problem 1: You seem to be thinking of ANOVA in the classical sense, probably specifically the analysis of balanced data. ANOVA and ANCOVA today are all performed with linear regression. It is just a matter of how you parameterize the model. You still have a linear predictor. Your linear regression model for ANCOVA is DV = constant + IV + CV + CV*IV + error. 

You might take a look at the extensive JMP Help about fitting linear models and the details of the factor parameterization.

Problem 2: The general approach is to fit the full ANCOVA model and then remove terms starting with the top of the hierarchy (i.e., term for interaction effect). This section of JMP Help about effect tests might help.

[#3]

Thank you for replying.

However, I can not get the point.

What I want to make clear is,...

Why can't I tell that "line is not same", when slope is heterogeneous.

I had tried to search that the rationales of above statement.

However, I couldn't, I could only found that try another method when homo-slope assumption is violated.

Why? Isn't ANCOVA's purpose compare multiple lines?

Then when not homo-slope, it is so obvious that lines are not same.

Please make me understand. please...

[#4]

Problem 1: Yes, you can use ANCOVA to fit a model to the continuous DP with terms for the continuous CV, the categorical IV and their interaction. The name 'ANCOVA' is historical and the original application was special case of a categorical factor (one-way ANOVA) and a continuous covariate to be simultaneously accounted for. We do not use terms like ANCOVA much anymore because the term 'linear predictor' is more general and includes the special case. As I said before, we tend to even think about ANOVA in a more general linear regression framework now, so we use regression to analyze models with categorical factors, with appropriate coding of the levels.

Problem 2: The first case (slope different, intercept different) is interpreted as a mean difference between groups and a proportional difference between groups. The interaction is important because the slope depends on the level of the categorical factor. The second case (slope different, intercept same) is interpreted as no mean difference between groups but there is a proportional difference between groups.

What is your context for these questions? Stability testing? What?

[#5]

Thank you for this continuous conversation.

I got the point from Problem 1, however can not get from 2.

It maybe caused by insufficient understanding in statistics.

I can not understand the "mean difference" and "proportional difference"

I had training course for industrial engineering which deals with statistics but by so far never heard of that word..., especially proportional difference.

what do you mean by mean difference?

is it right to understand that term as the case in "t-test" ?

in t-test, I compare mean of group1 and mean of group2...

By the way the purpose of the line comparison is,

to compare 'actual slope'.

pH meter, Spectrophotometer, The other many instrument that I use in the field use the <calibration curve> which is made by concentration-known standards.

Once made, concentration of interest can be measured and its concentration is calculated via <calibration curve>.

I supposed that every <cal. curve> should be same even though were made in different days.

As you inferred, right, they are not same at all.

So, I sorted them by, -repeat in day, -day, -operator, -lab, ...etc.

I want to see the minimum "ANCOVA SIGNIFICANCE" in left-most one, and the most "ancova significance" at right-most one. (you would understand that right-er one has more factor that contributing to variability)

However, I approached to hurdle that "homo-slope violation".

I really do not care about from what CV, it might possible to consider them as same.

I really do care that whether or not the slopes are SAME.

How much different? dont care.

So in this point, I desperately need your help.

Please.. !

[#6]

Let's see if we can sort out the answer to Problem 2 first. I am going to re-state the linear model to emphasize its interpretation.

What you fit: Y = b0 + categorical + continuous + categorical * continuous + error

What you interpret: Y = (b0 + categorical) + (1 + categorical) * continuous + error

The first sum is the constant in the regression model. If the categorical parameter in the first sum is significant, then it adds to b0, otherwise it is zero and adds nothing. If the categorical term is not significant, then there is only b0 or the common intercept. OTOH, if it is significant, then there is a unique or different intercept associated with each level of the categorical factor. Similarly, if the interaction term is significant, then it adds to the common slope to give a unique or different slope for each level of the categorical factor. Otherwise, it is zero and there is only the common slope.

Does this explanation help?

[#7]

I am soorry but not actually.

I can get the point of your last post.

But it can not tell that my problem 2.

"I really do care that whether or not the slopes are SAME."

Simple approach. (I don't need deep statistic theory in this case)

If significant, not same.

If not significant, same.

But for what criteria? slope? intercept?

There are only 2 cases that I become confused,

-slope in-significant and intercept significant

-slope significant and intercept in-significant.

and does this have rationales?  So far I can not found.

[#8]

Please not to give up to convince me.

I do not need solid rationales for my potential conclusion.

All I need to do is decide whether multiple line is considered to be same (or not).

You know, calibration (daily e.g.) curve supposed to be same, but by factors such as human-error contribute their 'inconsistency'.

I do want to decide whether they can be treated as 'same' or not.

Please help me. I need your help.

[#9]

I suggest that you look at Analysis > Specialized Modeling > Fit Curve. You supply the test method (Y), the standard method (X), and use an indicator column (Group). Select the Linear Fit command in the Polynomial group. The fit outline will have its own menu with equivalence and parallelism commands to assess similarity.

Do you test control samples with each calibration? Do you use control charts to evaluate consistency?

[#10]

attached is an example data table. Click green triangle next to the table script, "Pool Curves?)