Hi Akshatha, generally you can use mean substitution for missing data, but you'd have to ask your statistician whether it applies to this scenario, Good luck.

See https://www.researchgate.net/post/Two_way_ANOVA_using_missing_data_in_Minitab

Deborah J Hilton thank you for your answer. Please let me know how can we do mean substitution in Minitab. Kindly provide a stepwise information.

Hi Akshatha, I don't know MINITAB.  The book - SPSS Statistics -  A practical version 20. Allen P & Bennett K. pg 12-3 deals with missing values.  Kind Regards.

Sayed Ibrahim Ali

The different growth stages seem to be repeated measures, i.e. a within subjects factor). The cadmium concentrations are manipulated as a between subects factor. To test for the effects af stages and concentrations upon protein accumulations, you could use a non-parametric test such as the ARtTool or the f1.LD.f1 test from the nparLD R package or the .

For the ARTool, see: https://depts.washington.edu/acelab/proj/art/

or https://cran.r-project.org/web/packages/ARTool/readme/README.html

or: https://rcompanion.org/handbook/F_16.html

for the nparLD R package see:https://cran.r-project.org/web/packages/nparLD/nparLD.pdf

Thank you for response Jos Feys sir and Deborah J Hilton ma'am

If you have no data in the 200ppm stage 3 and 4 cells I don't think it makes sense to use any imputation method. I'd run the analysis excluding the 200ppm data but including the non-missing cells in a graphical or tabular summary. I don't think that the results of any imputation method would be very meaningful here.

If you have partial data in these cells then consider using multiple imputation (but never mean imputation) or similar.

Thom Baguley thank you for your response sir. If we exclude the data for 200 ppm Cd we get an error message saying that "The following terms cannot be estimated and were removed: STAGE*CADMIUM CONCENTRATION" in Minitab 19

Do you have complete data in the remaining cells? I'm not familiar with Minitab but if you exclude all 200ppm data (including for stages 0 to 2) it should run. You can't estimate this interaction if you don't have data in all cells of the design. If its repeated measures ANOVA commands also struggle with unbalanced cell sizes ... and a multilevel model would be better.

I agree with others who have suggested a multilevel model (aka., linear mixed model, or hierarchical linear model). I am not familiar with Minitab either, but it looks like you can estimate such a model.

https://support.minitab.com/en-us/minitab/20/help-and-how-to/statistical-modeling/anova/how-to/mixed-effects-model/before-you-start/overview/

Thom Baguley sir, I have attached my data below for your reference

I do not know plants, but if they have died is there any protein accumulation? If not, then these values are not missing, they are 0. Lots of techniques around if SURVIVAL is required to be in the data. Again, I do not know the issues about plants so can't specify what techniques might be appropriate in this context but I think commentators have latched onto the word MISSING without much explanation of what that means and if you know the mechanisms that lead to the data being "missing."

Thanks for providing the data, Akshatha Banadka. I changed it a bit to make it easier to use in any stats package, and I added a Plant ID number. See attached. Did I understand correctly that you have data for 18 plants? Thanks for clarifying.

Thank you Bruce Weaver sir . I could run it sucessfully however kindly let me know how can we assign letters to post hoc tukey paiwise comparison test

Akshatha Banadka, I don't understand what you mean about assigning letters for Tukey's test.

Meanwhile, let me ask again: Was I right in how I assigned the plant id numbers to 18 unique plants? Thanks for clarifying.

Yes Bruce Weaver sir, the assignment of plant id number is absolutely right.

Given that cdconc is a continuous variable, I'd go for a regression.

The relationship between log(Y) and cdconc looks quite linear, although not prefectly so. It might be better to use an appropriate non-linear function (logistic?) to better describe the relationship. However, the linearity is wite good and the assumption may not be too bad, given the ease of its interpretation.

It did a quick analysis in R using the package lmerTest (based on lme4; I just copied the text form R, the tables will be more readable when uing a monospaced font):

Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest'] Formula: log(Y) ~ cdconc * Stage + (1 | plantid) Data: x REML criterion at convergence: -118.1 Scaled residuals: Min 1Q Median 3Q Max -2.0958 -0.5059 -0.0699 0.5727 2.2735 Random effects: Groups Name Variance Std.Dev. plantid (Intercept) 0.001057 0.03252 Residual 0.002240 0.04733 Number of obs: 66, groups: plantid, 18 Fixed effects: Estimate Std. Error df t value Pr(>|t|) (Intercept) 1.6068612 0.0239939 43.4904881 66.970 < 2e-16 *** cdconc -0.0019118 0.0001981 43.4904881 -9.650 2.24e-12 *** Stage2 0.1181747 0.0279676 40.2022277 4.225 0.000133 *** Stage3 0.3107525 0.0292042 41.4054931 10.641 2.03e-13 *** Stage4 0.4692123 0.0292042 41.4054931 16.067 < 2e-16 *** cdconc:Stage2 -0.0005977 0.0002309 40.2022277 -2.588 0.013368 * cdconc:Stage3 -0.0001766 0.0002796 44.8480239 -0.632 0.530794 cdconc:Stage4 0.0001785 0.0002796 44.8480239 0.639 0.526371 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Single term deletions using Satterthwaite's method: Model: log(Y) ~ cdconc * Stage + (1 | plantid) Sum Sq Mean Sq NumDF DenDF F value Pr(>F) cdconc:Stage 0.02288 0.0076265 3 43.011 3.4049 0.02593 *

The interaction of Stage and cdconc is significant, meaning that there is sufficient data to conclude that the slopes of log(Y) against cdcons are not equal in all Stages, with Stage 2 having the most negative slope.

However, these differences might not be relevant either, so one might simplify the model further by excluding this interaction:

Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest'] Formula: log(Y) ~ cdconc + Stage + (1 | plantid) Data: x REML criterion at convergence: -153.3 Scaled residuals: Min 1Q Median 3Q Max -2.28910 -0.70502 0.00522 0.51184 2.28779 Random effects: Groups Name Variance Std.Dev. plantid (Intercept) 0.0007064 0.02658 Residual 0.0027037 0.05200 Number of obs: 66, groups: plantid, 18 Fixed effects: Estimate Std. Error df t value Pr(>|t|) (Intercept) 1.6285473 0.0195185 31.3883063 83.44 < 2e-16 *** cdconc -0.0021287 0.0001384 17.8472649 -15.38 9.69e-12 *** Stage2 0.0584067 0.0173324 44.8332505 3.37 0.00156 ** Stage3 0.2903020 0.0184096 46.2846764 15.77 < 2e-16 *** Stage4 0.4771760 0.0184096 46.2846764 25.92 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Single term deletions using Satterthwaite's method: Model: log(Y) ~ cdconc + Stage + (1 | plantid) Sum Sq Mean Sq NumDF DenDF F value Pr(>F) cdconc 0.63967 0.63967 1 17.847 236.59 9.688e-12 *** Stage 2.24927 0.74976 3 45.766 277.31 < 2.2e-16 ***

Obviousely, the simple effects of both, cdconc and Stage, are statistically significant. The response decreases with cdconc and very clearly increases with Stage.

Thank you so much Dr. Jochen Wilhelm for your descriptive and clear answer. I would try that.

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