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R lm repeated measures

Repeated Measures in R Mar 11th, 2013 In this tutorial, I'll cover how to analyze repeated-measures designs using 1) multilevel modeling using the lme package and 2) using Wilcox's Robust Statistics package (see Wilcox, 2012). In a repeated-measures design, each participant provides data at multiple time points Repeated measures analysis with R Summary for experienced R users The lmer function from the lme4 package has a syntax like lm. Add something like + (1|subject) to the model for the random subject effect

Repeated Measures in R - Jason A

We will begin with the two-level model, where we have repeated measures on individuals in different treatment groups. Unconditional model. Model formulation. Level 1 Y i j Level 2 β 0 j = β 0 j + R i j = γ 0 0 + U 0 j with, U 0 j ∼ N (0, τ 0 0 2 ), and. R i j ∼ N (0, σ 2) To fit this model we ru Repeated Measures and Mixed Models - Michael Clar Repeated Measures Analysis with R There are a number of situations that can arise when the analysis includes between groups effects as well as within subject effects. We start by showing 4 example analyses using measurements of depression over 3 time points broken down by 2 treatment groups

Kickstarting R - Repeated measures

In the approach here we will use a repeated measures analysis with all the measurements, treating Student as a random variable to take into account native differences among students, and including an autocorrelation structure. Input = (Instruction Student Month Calories.per.day 'Curriculum A' a 1 2000 'Curriculum A' a 2 1978 'Curriculum A' a 3 1962 'Curriculum A' a 4 1873 'Curriculum A' a 5. Dear Ingo, One approach would be to use the Anova() function in the car package. See ?Anova and in particular the O'Brien and Kaiser example, which is for a more complicated repeated-measures design. If you want to get type-III tests (as opposed to the default type-II tests), be careful with the contrast coding for the between-subjects factors Originally I was going to do a repeated measures ANOVA, but 5 out of the 11 have one missing time point, so linear mixed model was suggested so I don't lose so much data. I use R primarily and was. Repeated Measures in R One Factor Reported Measures. First, we will look at the example done in class from the book. Six judges are used, each judging four wines. Again, treat the judges as blocks. In theory, the order in which the judges taste the wine should be random

Repeated measures data consist of measurements of a response (and, perhaps, some covariates) on several experimental (or observational) units. Frequently the experimental (observational) unit is Subject and we will refer to these units as\subjects. However, the methods described here are not restricted to data on human subjects. Longitudinal data are repeated measures data in which the. Another common set of experiments where linear mixed-effects models are used is repeated measures where time provide an additional source of correlation between measures. For these models we do not need to worry about the assumptions from previous models, since these are very robust against all of them. For example, for unbalanced design with blocking, probably these methods should be used. Multi-level Models and Repeated Measures Between schools 0.166 Between students 3.219 This is misleading. From our earlier investigation, it is clear that the difference is between classes, not between schools! 10.2.3 Predictive accuracy 10.3 A Multi-level Experimental Design 10.3.1 The anova table 10.3.2 Expected values of mean squares 10.3.3∗ The sums of squares breakdown 10.3.4 The. Repeated measures design is a research design that involves multiple measures of the same variable taken on the same or matched subjects either under different conditions or over two or more time periods. For instance, repeated measurements are collected in a longitudinal study in which change over time is assessed

r - Using lmer for repeated-measures linear mixed-effect

  1. GLM repeated measure is a statistical technique that takes a dependent, or criterion variable, measured as correlated, non-independent data. Commonly used when measuring the effect of a treatment at different time points. The independent variables may be categorical or continuous. GLM repeated measure can be used to test the main effects within and between the subjects, interaction effects.
  2. g a repeated measures analysis of variance. PROC GLM provides both univariate and multivariate tests for repeated measures for one response
  3. That is, not paired or repeated measures data • Moderate deviation from normally-distributed residuals is permissible . Hypotheses • Null hypothesis: The means of the measurement variable for each group are equal. • Alternative hypothesis (two-sided): The means of the measurement variable for among groups are not equal. Interpretation • Reporting significant results for the omnibus.
Effect of transaortic constriction (TAC) on LV structure

r - repeated measure anova using regression models (LM

Repeated measures are those measures that are repeated on more than one occasion, we met an example in the paired t statistic chapter. Taking another example, say we have cholesterol levels measured four times (t 1, t 2, t 3, t 4) over a year for each patient following commencement of a new cholesterol lowering drug. So for each patient we have four measurements which can be depicted thus: We. Course Description. This course focuses on within-groups comparisons and repeated measures design. With the help of a working memory training experiment, one of Professor Conway's main areas of research, it will be explained what the pros and cons are of a repeated measures design and how to conduct the calculations in R yourself Title: Microsoft PowerPoint - AERA2007_overheads.ppt Author: jkrobert Created Date: 4/5/2007 5:11:29 P Approach 1: Repeated Measures Multivariate ANOVA/GLM. When most researchers think of repeated measures, they think ANOVA. In my personal experience, repeated measures designs are usually taught in ANOVA classes, and this is how it is taught. The data is set up with one row per individual, so individual is the focus of the unit of analysis. This is called the wide format. The multiple measures. Two-way repeated-measures ANOVA; Two-way split-plot ANOVA; Three-way split-plot ANOVA; Mixed effects models; Sum of squares type I, II, and III; General Topics; Assess normality; Assess variance homogeneity; Nonparametric . Overview; Classical nonparametric methods; Location tests for one and two samples (Sign, Wilcoxon signed-rank, Wilcoxon rank-sum / Mann-Whitney-U) Location tests for more.

Assumptions of MANOVA. MANOVA can be used in certain conditions: The dependent variables should be normally distribute within groups. The R function mshapiro.test( )[in the mvnormtest package] can be used to perform the Shapiro-Wilk test for multivariate normality. This is useful in the case of MANOVA, which assumes multivariate normality.. Homogeneity of variances across the range of predictors The underlying least squares arithmetic of aov and lm is identical. In R, the QR algorithm is used. The difference between the two is intent of the analysis and the default presentation of the results. With lm [Linear Model], the focus is on the effect of the individual columns of the predictor matrix. The columns are usually interpreted as values of real-valued observations

R Tutorial Series: Two-Way Repeated Measures ANOVA R

Using R and lme/lmer to fit different two- and three-level

  1. ant ways to fit multilevel models that account for such structure in.
  2. Likewise, in repeated measures data, individuals typically display a high degree of similarity in responses over time. Non-independence may be considered either a nuisance variable or something to be substantively understood, but the prevalence of nested data requires that analysts have a variety of tools to deal with nested data. The term multilevel analysis is frequently used to.
  3. The r package simr allows users to calculate power for generalized linear mixed models from the lme 4 package. The power calculations are based on Monte Carlo simulations. It includes tools for (i) running a power analysis for a given model and design; and (ii) calculating power curves to assess trade‐offs between power and sample size
  4. The linked Dropbox file has code and data files for doing contrasts and ANOVA in R. https://www.dropbox.com/sh/132z6stjuaapn4c/AAB8TZoNIck5FH395vRpDY..
  5. Course Description. Analysis of Variance (ANOVA) is probably one of the most popular and commonly used statistical procedures. In this course, Professor Conway will cover the essentials of ANOVA such as one-way between groups ANOVA, post-hoc tests, and repeated measures ANOVA
  6. data and repeated measures. The GEE approach focuses on models for the mean of the correlated observations within clusters without fully specifying the joint distribution of the observations. It has been widely used in statistical practice. This paper illustrates the application of the GEE approach with geepack through an example of clustered binary data. Keywords: generalized estimating.
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Anova Tables for Various Statistical Models. Calculates type-II or type-III analysis-of-variance tables for model objects produced by lm, glm, multinom (in the nnet package), polr (in the MASS package), coxph (in the survival package), coxme (in the coxme pckage), svyglm (in the survey package), rlm (in the MASS package), lmer in the lme4 package, lme in the nlme package, and (by the default. Version info: Code for this page was tested in R version 3.1.2 (2014-10-31) On: 2015-06-15 With: knitr 1.8; Kendall 2.2; multcomp 1.3-8; TH.data 1.0-5; survival 2.37-7; mvtnorm 1.0-1 After fitting a model with categorical predictors, especially interacted categorical predictors, one may wish to compare different levels of the variables than those presented in the table of coefficients Repeated measure anova There are several predictive variables, say X1 and X2, and several observations for each value of (X1,X2). This is a particular case of two-way anova -- a case in which we can, if needed, consider an interaction term: the model would be y~x1+x2 (no interaction) of y~x1*x2 (interaction). Cross-factor anov allow for the analysis of non-normal, clustered data (e.g. repeated measures, Littel et al. 2002). These models are usually concerned with population-level inferences. Finally, Generalized Linear Mixed Models (GLMMs) have been developed more recently and extend GLMs to include random-effects (Agresti 2002). In contrast with GEEs, these models often have subject- specific interpretations. This.

method (Moder, 2007; Hotteling, 1931). 2 on page 576 explains the ANOVA table for repeated measures in both factors. It appears that SAS uses an awfully strange definition of what is the type III sums of squares method. R is consistent. The normalized ratios σ 2 u(f) σ2(f) serve as indicators for the importance of the variable ξ. The anova and aov Functions in R. values and residuals. Learn more about repeated measures analysis using mixed models in our most popular workshop, Analyzing Repeated Measures Data: GLM and Mixed Models Approaches. I wanted to share this recent publication with you. I would not have made it through the statistical analysis for this project without your Repeated Measures class. I am continually pulling out my class notes for other analyses. Thanks. Performance Measures: Three sets of measurements are provided. Residual Standard Error: This is the standard deviation of the residuals. Smaller is better. Multiple / Adjusted R-Square: For one variable, the distinction doesn't really matter. R-squared shows the amount of variance explained by the model. Adjusted R-Square takes into account. This repeated measures design allows the anticipated variability in stress tolerances between individual sharks to be accounted for in the analysis (so as to permit more powerful test of the main treatments). Furthermore, by performing repeated measures on the same subjects, repeated measures designs reduce the number of subjects required for the investigation. Essentially, this is a. Jeff didn't actually say anything to restrict us to standardized mean difference type measures (as opposed to, say, variance explained type measures), and we can only guess whether the correct effect size he has in mind is a d-like measure or an -like measure or what. But here I'll focus on d-like measures, which are perhaps the most popular for studies with categorical predictors.

Repeated Measures and Mixed Models - Michael Clar

  1. Three-way Anova with R Goal: Find which factors influence a quantitative continuous variable, taking into account their possible interactions stats package - No install required Y ~ A + B Plot the mean of Y for the different factors level
  2. 1) One might observe a clear difference between k-fold and repeated k-fold cross-validation with a large data set with thousands of rows. Here we have only 47 rows in the swiss data set. The results obtained with the repeated k-fold cross-validation is expected to be less biased compared to a single k-fold cross-validation
  3. MEMORE (MEdiation and MOderation in REpeated-measures designs)Montoya, A. K., & Hayes, A. F. (2017). Two condition within-participant statistical mediation analysis: A path-analytic framework. Psychological Methods, 22, 6-27. [MEMORE is a macro for SPSS and SAS that estimates the total, direct, and indirect effects of X on Y through one or more mediators M in the two-condition or two-occasion.
  4. g the responses using a suitable within-subjects model matrix, for example extending the linear hypothesis in Equation2to H0: L (q p) B (p m) P (m v) = C (q v) (3) Here, the response-transformation matrix P, assumed to be of full column-rank, provides contrasts in the responses (see, e.g.,Hand and.
  5. In der Statistik ist das allgemeine lineare Modell (ALM bzw. englisch general linear model, kurz: GLM), auch multivariates lineares Modell (englisch multivariate linear model) ein lineares Modell, bei der die abhängige Variable kein Skalar, sondern ein Vektor ist. In diesem Fall wird ebenfalls konditionierte Linearität ⁡ (∣) = wie beim klassischen Modell der linearen Mehrfachregression.

Repeated Measures Analysis with R - IDRE Stat

Repeated k-fold Cross Validation. The process of splitting the data into k-folds can be repeated a number of times, this is called Repeated k-fold Cross Validation. The final model accuracy is taken as the mean from the number of repeats. The following example uses 10-fold cross validation with 3 repeats to estimate Naive Bayes on the iris dataset My second comment is another question, on the random intercept and slope for repeated measures you include on model lme2 the variable year, why? don't you want only the random effect? this way you get also a fixed effect for year, don't you? Thank you for your post and in advance for your help. Helena . Reply Delete. Replies. Fabio Veronesi 4 July 2017 at 10:23. Hi Helena, The first question. Repeating things: looping and the apply family. 18 March 2013. Previously we looked at how you can use functions to simplify your code.Ideally you have a function that performs a single operation, and now you want to use it many times to do the same operation on lots of different data This PR is for adding repeated measures ANOVA Tested up to three factors and it reproduces R ezANOVA R Code for producing the results in test_two_factors_repeated. Here we see that linearity seems to hold reasonably well, as the red line is close to the dashed line. We can also note the heteroskedasticity: as we move to the right on the x-axis, the spread of the residuals seems to be increasing

R Handbook: Repeated Measures ANOV

Each measurement is uniquely identified by a combination of ID variables (in this case ID, Time, and whether the measurement is on X1 or X2). For example, the measured value 5 in the first row is uniquely identified by knowing that it's from observation (ID) 1, at Time 1, and on variable X1. Melting. When you melt a dataset, you restructure it into a format where each measured variable is in. I was recently asked about when to use one and two tailed tests. The long answer is: Use one tailed tests when you have a specific hypothesis about the direction of your relationship. Some examples include you hypothesize that one group mean is larger than the other; you hypothesize that the correlation is positive; you [

matrix, X, in the same way that the R lm function generates model matrices. However, a mixed-model formula incorporates k≥1 random-effects terms of the form (r|f) as well. These kterms are used to produce the random effects model matrix, Z(Equation2; Sec Intro. Panel data (also known as longitudinal or cross -sectional time-series data) is a dataset in which the behavior of entities are observed across time We use Regression analysis to create models which describe the effect of variation in predictor variables on the response variable. Sometimes, if we have a categorical variable with values like Yes/No or Male/Female etc. The simple regression analysis gives multiple results for each value of the. # Convert to long format library (reshape2) dfw_long <-melt (dfw, id.vars = subject, measure.vars = c (pretest, posttest), variable.name = condition) dfw_long #> subject condition value #> 1 1 pretest 59.4 #> 2 2 pretest 46.4 #> 3 3 pretest 46.0 #> 4 4 pretest 49.0 #> 5 5 pretest 32.5 #> 6 6 pretest 45.2 #> 7 7 pretest 60.3 #> 8 8 pretest 54.3 #> 9 9 pretest 45.4 #> 10 10 pretest 38.9.

Two-way ANOVA in R statstutor Community Project © Sofia Maria Karadimitriou and Ellen Marshall www.statstutor.ac.uk University of Sheffield stcp-karadimitriou-ANOVA Click for the complete R source code for this tutorial. Note that this is automatically generated (tangled) from the org mode source file for this document, which adds some extra commands to specify filenames for plots (and to subsequently close the graphics device).Note that if you use 'source' to read in the R code, the ggplot2 plots will not be created as auto-printing is turned off when. I have attended the useR! conferences every year now for the past 9 years, and loved it! However, this year I'm saddened that I won't be able to go. This is because this year the conference will be held in Australia, and going there would require me to be away from home for at least 8 days (my heart goes to the people of Australia who had a hard time coming to useR all these years) perform two-way repeated measures anova in python using {car} package in R, via rpy2 - anova.py. Skip to content. All gists Back to GitHub. Sign in Sign up Instantly share code, notes, and snippets. d1manson / anova.py. Last active Sep 22, 2017. Star 4 Fork 0; Code Revisions 2 Stars 4. Embed. What would you like to do? Embed Embed this gist in your website. Share Copy sharable link for this.

R help - Repeated Measures Analysis - GL

  1. Details. Models for lm are specified symbolically. A typical model has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a linear predictor for response.A terms specification of the form first + second indicates all the terms in first together with all the terms in second with duplicates removed
  2. t-tests. The t.test( ) function produces a variety of t-tests. Unlike most statistical packages, the default assumes unequal variance and applies the Welsh df modification.# independent 2-group t-test t.test(y~x) # where y is numeric and x is a binary facto
  3. It measures the fit when a penalty is applied to the number of parameters. Smaller AIC values indicate the model is closer to the truth. Null deviance: Fits the model only with the intercept. The degree of freedom is n-1. We can interpret it as a Chi-square value (fitted value different from the actual value hypothesis testing). Residual Deviance: Model with all the variables. It is also.
  4. Using RLM in repeated measures generates regression coefficients with standard errors biased. This requires covariance matrix application that will produce more reliable estimates, in others words, narrower confidence intervals from Mixed Effects Models.4 This is the best alternative to verify changes over time or the conditioners effects on repeated measures outcomes in longitudinal studies.
  5. Begriff Zeitreihe. Eine Zeitreihe ist eine zeitlich geordnete Folge (meist aber keine Reihe im mathematischen Sinne) von Beobachtungen einer bestimmten Größe. Die einzelnen Zeitpunkte werden zu einer Menge von Beobachtungszeitpunkten zusammengefasst, bei der für jeden Zeitpunkt ∈ genau eine Beobachtung vorliegt. Zeitreihen treten in allen Bereichen der Wissenschaft auf
  6. An Example of ANOVA using R by EV Nordheim, MK Clayton & BS Yandell, November 11, 2003 In class we handed out An Example of ANOVA. Below we redo the example using R
  7. Traditional repeated measures Anova; Comparison with a multilevel model; Checking assumptions; Followup tests; 9 Generalized linear models. Logistic regression; 10 Multilevel models. Fitting multilevel models in R. Use lmer and glmer; p values in multilevel models; Extending traditional RM Anova. Fit a simple slope for Days; Allow the effect of sleep deprivation to vary for different particip

Can anyone suggest some R script for repeated measures GLM

Also called the coefficient of determination, this is an oft-cited measurement of how well your model fits to the data. While there are many issues with using it alone (see Anscombe's quartet ) , it's a quick and pre-computed check for your model Next message: John Fox: Re: [R] MANOVA with repeated measurements Previous message: John Kane: Re: [R] changing existing graphs optimally In reply to Dimitri Szerman: [R] lm() intercept at the end, rather than at the beginning Next in thread: Greg Snow: Re: [R] lm() intercept at the end, rather than at the beginning Contemporary messages sorted: [ By Date] [ By Thread] [ By Subject. repeated-measures analysis. In each case, there is an extraneous feature of the research design that makes each observation more related to others than what would be prescribed by the model. For example, if one is conducting a study of mathematics test scores in a specific school, students taking classes in the same classroom or being taught by the same teacher would very likely produce scores. Logistic regression in R Programming is a classification algorithm used to find the probability of event success and event failure. Logistic regression is used when the dependent variable is binary(0/1, True/False, Yes/No) in nature. Logit function is used as a link function in a binomial distribution In R, using lm() is a special case of glm(). lm() fits models following the form Y = Xb + e, where e is Normal (0 , s^2). glm() fits models following the form f(Y) = Xb + e. However, in glm both the function f(Y) (the 'link function') and the dist..

For linear relationships we can perform a simple linear regression.For other relationships we can try fitting a curve. From Wikipedia:. Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints Needs for blocking, repeated measurements or replication may lead to modifications of the design. Example: Car seat occupation run.no run.no.std.rp Day Foam Bolster Aged D E F. 1 1 8.1.8 1 soft stiff aged + + - 2 2 4.1.4 1 soft soft new + + - 3 3 12.1.12 1 hard soft aged + + - 4 4 15.1.15 1 hard stiff new + - +. Leverage is a measure of how much each data point influences the regression. The plot also contours values of Cook's distance, which reflects how much the fitted values would change if a point was deleted. A point far from the centroid with a large residual can severely distort the regression. For a good regression model, the red smoothed line should stay close to the mid-line and no point. When single sampling units are selected amongst highly heterogeneous conditions, it is unlikely that these single units will adequately represent the populations and repeated sampling is likely to yield very different outcomes. In the depiction of a single factor ANOVA below, a single quadrat has been randomly placed in each of the six Sites.

R/lmGroup.R defines the following functions Learn how generalized linear models are fit using the glm() function. This covers logistic regression, poisson regression, and survival analysis By Andrie de Vries, Joris Meys . Before you can use R's aov() function with your data, you'd better set the contrasts you're going to use. Contrasts are very often forgotten about when doing ANOVA (analysis of variables), but they generally help with interpreting the model and increase the accuracy of aov() and the helper functions

Repeated Measures in R - Pennsylvania State Universit

> repeated measures for each of the output variables to include > intraobserver variability in the design. > > How can I perform such a repeated-measures manova using R? I > think I have found how to perform the non-repeated version of > this experiment based on previous posts, but I am having > trouble generalizing to the repeated-measures version (maybe > I am just too tired today. r . Lm = (~l) i ( 4.> r ) 2m - ( <P '( ) 1m) , b. ij, = f)jlm-f. ij2 ( 4 )q . Equation (4) is the model for the t-time completely randomized repeated measurements design. The carryover effects and the sequence and time effects are evaluated in terms of . Sijm! sums of the observations at each time point, substitute the expression (5) into equation (1), where (Xij = B. ijm = + + e. ij2 + f.

Appearance. The Really Long Meter looks like a usual meter, but is way longer. Its length varies between one really long meter (in short: rlm) and one really long meter.Because of this huge difference it is hard to define, how long a rlm is; thus in many cases it is just repeated what was repeated before Most stats packages present two R squared measures. In In repeated samples, the R squared estimates will be above 0, and their average will therefore be above 0. Since the bias is the difference between the average of the estimates in repeated samples and the true value (0 here), the simple R squared estimator must be positively biased. Incidentally, lots of estimators we used are not.

The repeated-measures transformation matrix consists of the columns of the intra-subject model matrix corresponding to the term or terms in iterms. In most instances, this will be the simpler approach, and indeed, most tests of interests can be generated automatically via the Anova function Longitudinal Data and Correlated Measures Bias: The Alternative of Mixed Model Two-Way Repeated Measures ANOVA; Two-Way ANOVA with Unequal Sample Sizes; ANOVA Pairwise Comparison Methods; Reshape Package for ANOVA Data; Factor Analysis. Exploratory Factor Analysis; News, Updates, & Other Topics. Statistical Analysis with R Released; Statistical Analysis with R Book Reviews; Citing R with EndNote ; License. R Tutorial Series by John M. Quick is governed under a Creative. 2 3-Way Factorial Designs There are 7 effects involved in a 3-way factorial • 3 main effects (one for each IV) • 3 2-way interactions (one for each pair of IVs R/LM/25:2015 Procedure for recognizing technical devices as measuring instruments Approved measurements made by the technical devices under review provide accuracy that is capable of being assessed against requirements of applicable laws and technical regulations, as well as local regulations published by the state administration bodies of the respective COOMET member country; c) the.

Note: the lmer function in the lme4 package also can be used for the same results. We will be using the lme4 package later in the course. For both the SPSS analysis and the R analysis, a one-tailed test should be use Repeated measures ANOVA *adjusted P < 0.05, Dunnett correction for multiple comparisons. (C) Percentage of CFSE low T cells from dBT co-culture experiments described (n = 7). Repeated measures ANOVA *adjusted P < 0.05 Dunnett correction for multiple comparisons. PD-L1 Expression Is Important for the Induction of NCMs but Not MDSCs. Having demonstrated that GBM-EVs induce monocytes to.

r - Unable to run Two-way repeated measures ANOVA; 0 (non12 GEE, Continuous Outcome: Beat the Blues | EncyclopediaWHymes_Hyena Presentation_ABRCMSImmunofluorescence quantification of TSC2 (Red) and WGAComputational models of O-LM cells are recruited by low orVenezuela arrests two journalists, German envoy expelled

3.1.2.2. Paired tests: repeated measurements on the same individuals ¶ PIQ, VIQ, and FSIQ give 3 measures of IQ. Let us test if FISQ and PIQ are significantly different. We can use a 2 sample test: >>> 14.7 Repeated measures ANOVA using the lme4 package; 14.8 Test your R might! 15 Regression. 15.1 The Linear Model; 15.2 Linear regression with lm() 15.2.1 Estimating the value of diamonds with lm() 15.2.2 Getting model fits with fitted.values; 15.2.3 Using predict() to predict new data from a model; 15.2.4 Including interactions in models: y. Between 1933 and 1945, the organization of the Luftwaffe underwent several changes. Originally, the German military high command, for their air warfare forces, decided to use an organizational structure similar to the army and navy, treating the aviation branch as a strategic weapon of war Pharmacological analysis was based on microarray data and performed by first fitting a mixed-effect, repeated measures model within each TC separately. The response variable was log2-expression. Main effects, including treatment and time along with their interactions, were included in the model as fixed discrete effects. Age, sex and cell counts were included in the model as additional control.

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