I have been working on a multivariate linear mixed regression analysis in JMP and would like to check in to determine: QUESTION 1. whether or not I am on the right track?
Data collected for the analysis described below stems from a field experiment possessing a randomized complete block design, with 4 replications. The experiment has an additive design, in which crop density is elevated incrementally (200, 300, 400, and 500 plants m^-2) across two row spacings (12 and 20 cm). I am interested in evaluating how crop biomass (g m^2) is affected by the continuous variable crop density (observed values ranging from 109 to 494 plants m^-2) and the two levels of the nominal variable, row spacing (15 and 20 cm).
Using the fit model platform, barley biomass is evaluated as a function of row spacing (nominal) crop density (continuous), and an interaction between row spacing * crop density. Block is also included as a random effect.
Non-significant variables are removed from the model; in this case, the interaction term between row spacing * crop density was removed.
After checking the residuals by predicted plot I select dropdown > save columns > prediction forumula.
Here is what is reported in the reduced 'fit model' platform:
And when I click on the "+" symbol in the columns bar of my data table to view the 'prediction formula' saved, this is what it looks like:
I have also made a figure in graph builder to go along with this analysis. Lines are fit to = the predicted formula on the y-axis * the continuous variable crop density on the x-axis; points are fit to the continuous variable barley biomass on the y-axis * average crop density within plots receiving the 200, 300, 400, and 500 planting densities (hence the 4X dots) across blocks by row spacing.
I would like to report in a table showing the parameter estimates and associated standard errors for the relationship between crop density (plants m-2) and all crop-biomass (g m-2), f1 (x), evaluated using the linear mixed model:
f1(x) = a(r) + b(r) * d r = 1,2
where a represents the crop biomass when crop density equals zero, b is the slope as crop density increases, d is crop density, and r1 and r2 are RS15 and RS20, respectively.
The slope, parameter b is the same for both row spacings and is 0.8479 (as seen in the predicted formula, and the parameter estimates report), and the standard error is 0.2066 (as seen in the parameter estimates report).
To calculate the intercept for the row spacing of 15 cm, parameter a(r1), I calculate 1436 - 233 = 1203 (pulling numbers from the predicted formula) and pull the standard error from the parameter estimates report, which is 19. The intercept for row spacing 20 cm, parameter a(r2), is therefore 1436 + 233 = 1669, and now I am guessing that also the standard error in this case is also 19?
Therefore, like when the 'equation' box is checked in the graph builder platform under 'line of fit' Y (15) = 1203 + 0.8479 * X and Y (20) = 1670 + 0.8479 * X.
QUESTION 2. Is this the correct way to calculate/obtain parameter estimates and their standard errors for reporting?
Here is a second example where the interaction term row spacing * crop density is significant and therefore keep in the model.
Here are the parameter estimates from the fit model report:
Here is the prediction formula from the saved column:
And here is the figure:
QUESTION 3. How do I go about extracting parameter estimates and standard errors in this instance?
It looks like you have done some good work here.
I am sorry but I find it hard to say whether you are "doing this right" just from the screenshots. It does not give me the full picture.
It can be quite complicated when you have a mixed model as you are saying that there are different components of the variation. Reporting the standard error is more complex because of this.
Are you able to attach the data table? Or an anonymised version?
Sure thing, and thank you for your feedback.
I have attached a 'reduced' JMP data set containing elements applicable to the two examples given in my initial post.
A quick 'tour' of items included:
- Model saved to data table titled: "CROP EFFECT ANALYSIS" is the model I used to perform multivariate mixed linear regression performed above for response variables with column titles "MP_BarleyBiomass_GM2" and "SP_BarleyBiomass_GM2".
- Additional scripts saved to the data table, titled, "GRAPH1_MP_BarleyBiomass" and "GRAPH2_SP_BarleyBiomass" will produce the figures presented in my initial post visualizing results.
- The column titled, "TRT_RowSpacing_Cm" is a nominal explanatory variable used in the model.
- "TRT_CropDensity_NoM2" is the target crop density (i.e., the treatment) prescribed to each plot, 200, 300, 400, or 500 plants m^-2, while "FP_CropDensity_NoM2" is the achieved crop density measured for each plot. FP_CropDensity_NoM2 is used as a continuous variable in the model.
- Finally, the column titled "ACHIEVED_CropDensity_NoM2_AvgAcrossBlocksByTargetCropDensityByRowSpacing" = acheived crop density (FP_CropDensity_NoM2) averaged across plots having the same target crop density (TRT_CropDensity_NoM2) and row spacing (TRT_RowSpacing_Cm) this column is used for graph building purposes.
Please let me know if you have any additional questions.
How to identify parameter estimates (especially for the SP_BarleyBiomass_GM2 example shown in my previous post) and their corresponding standard errors are my foremost question I can't quite figure out at this point.
Best wishes,
Margaret
Thanks, Margaret. That really helps.
I am interested in why you have a block effect. It seems like the observations (rows) in each block are from separate plots and so they are completely independent observations of the same treatment. But you know more about the experiment than me so you probably know something I have missed.
In terms of reporting the parameter estimates, you have everything you need in the parameter estimates table and I hope you can see how they relate to the Prediction Expression in JMP and the equation form that you wish to use.
However, you need to know that when you have a nominal effect with 2 levels in a model in JMP, the intercept is the prediction of the response for the average over those 2 levels. But, if you need to, you can easily calculate the intercept for each level of the nominal...
For your example the intercept is (approximately) 1436 and the effect for row spacing = 15 is -233.
The intercept for row spacing = 15 is therefore 1436 - 233 = 1203.
The intercept for row spacing = 20 is therefore 1436 + 233 = 1669.
The standard errors are the same regardless of how you define the intercept.
I recommend that you use the Prediction Profiler (red triangle menu at the top of the fit model report > Factor Profiling > Profiler). I think this is the best way to understand your model. You can set the variables to any value and see the response predicted by the model. For example, here you can clearly see the intercept (when crop density = 0) is 1203 with row spacing = 15:
And you can see how increasing crop density by 1 unit increases the predicted response by 0.85 units:
Similarly I think this should help you to understand how the interaction effect is parameterised.
I hope this helps.
Phil
