In this vignette, we want to discuss how to specify multilevel multivariate models usingbrms. we call it a modelmultivariateif it contains multiple response variables, each predicted by its own set of predictors. Consider an example from biology. Hadfield, Nutall, Osorio, and Owens (2007) analyzed data from the Eurasian blue tit (https://en.wikipedia.org/wiki/Eurasian_blue_tit). They predicted it
hocklength, as well as the
to returncolor of chicks. Half of the litter was placed in another
nest of adoption, while the other half stayed in the nest of their own children
barrier. This makes it possible to separate genetic from environmental factors. In addition, we have information about
Date of birthm
sexof offspring (the latter are known in 94% of animals).
data("BT Data",package = "MCMCglmm")head(BT data)
tarsus posterior animal mother son foster hatchery sex1 -1.89229718 1.1464212 R187142 R187557 F2102 -0.6874021 female2 1.13610981 -0.7596521 R1189725 0 .9 8468946 0.14 49373 R187341 R187568 A602 -0.4279814 Male4 0.37900806 0.2555847 R046169 R187518 A11465 -1 07525299 -0.3006992 R04 6161 R187528 A2602 -1.4656641 Female6 -1.13519543 1.5577219 R187409 R187945 C2302805 Female 0.3
Basic multivariate models
We start with a relatively simple multivariate normal model.
bform1<- friend(mvbind(tarsus, Kostas)~sex+Date of birth+(1|Pi|adoption nest)+(1|q|barrier))+ set_rescor(TRUE)setting 1<- brm(bform1,data =BT data,chains = 2,cores = 2)
As can be seen from the template code, we use
mvbindnotation for measurementbrmsthat both
to returnare separate response variables. the term
(1|p|first child)indicates an interrupt variable
nest of adoption. Written
|p|in the middle, we indicate that all variable effects of
nest of adoptionthey must be modeled as correlated. This makes sense, as we actually have two parts of the model, one for
hockand one for
to return. the index
Piis arbitrary and can be replaced by other symbols that come to mind (for details on the multi-level syntax ofbrms, ed
βινιέτα ("brms_multililevel")). Likewise, the term
(1|q|dam)indicates correlated maternal genetic variable effects of neonates. Alternatively, we could also have modeled genetic similarities through pedigrees and matching paternity matrices, but this is not the focus of this vignette (see
βινιέτα ("brms_phylogenetics")). The results of the model are easily summarized
setting 1<- add_criterion(adjustment 1,"toilet")summary(adaptation 1)
Family: MV (Gaussian, Gaussian) Links: mu = identity; sigma = identity mu = identity; sigma = identity Type: tarsus ~ sex + date of birth + (1 | p | foster nest) + (1 | q | mother) back ~ sex + date of birth + (1 | p | foster nest) + (1 | q | mother ) Data: BTdata (Number of observations: 828) Plots: 2 strings, each with iter = 2000; heating = 1000; skinny = 1; Total Bonds After Heat = 2000 Group Level Effects: ~ dam (number of levels: 106) Rating Rating. Error l-95% CI u-95% CI Rhat Bulk_ESSsd(tarsus_Intercept) 0.48 0.05 0.39 0.59 1.00000 back 0.10 0.39 1.01 253cor (tarsus_Intercept,back_Intercept) -0, 51 0.21 -0.91 -0.07 1.00 468 Tail_ESSsd(tarsus_Intercept) 1301sd(back_Intercept,back_Intercept,back_Intercepttar) (back_Intercepttar) of Levels: 104 ) ErrorEstimateEst l-95% CI u -95% CI Rhat Bulk_ESSsd (tarsus_Intercept) 0.27 0.06 0 .17 0.38 1.00 526sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 441corp (back_Tarsus_Intercept) 0.35 0.06 0.23 0.47 1.00 441 cor. .9 9 1.04 136 Tail_ESSsd( tarsus_Intercept) 862sd(back_Intercept) 947cor(tarsus_Intercept, back_Intercept ) 517Population Level: Estimate Estimate Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESStarsus_Intercept -0.41 0.07 -0.55 - 0.27 1 .00 1303_13001.601. 1.00 2243 16 68tarsus_sexMale 0.77 0.06 0.65 0.88 1.00 4070 1440tarsus_sexUNK 0.23 0.13 -0.03 0.48 1.00 3196 1554tarsus_hatchdate -0.04 0.06 -0. ,16 0.07 1.00 105001. 0.14 1.00 4881 1422back_sexUNK 0.1 5 0.14 -0.12 0.42 1.00 3668 1653back_hatchdate -0.09 0.05 -0.19 0. 01 1.00 178F specifically : Estimate Estimate Error l- 95% CI u-95 % CI Rhat Bulk_ESS Tail_ESSsigma_tarsus 0.76 0.02 0.72 0.80 1.00 2758 1442sigma_back 0.90 0.02 0.010.501. ations: Estimate Est.Error l-95 % CI u-95% CI Rhat Bulk_ESS Tail_ESSrescor(tarsal,back) - 0.05 0.04 -0.13 0.02 1.00 3343 1663 Draws were sampled by sampling (NUTS ). For each parameter, Bulk_ESS and Tail_ESS are effective measures of sample size, and Rhat is the potential downscaling factor in segregation chains (at convergence, Rhat = 1).
The summary output of multivariate models is very similar to univariate models, except that the parameters are now prefixed with the corresponding response variable. In mothers, tarsus length and back color appear to be negatively correlated, while in foster nestlings the opposite occurs. This indicates differential effects of genetic and environmental factors on these two traits. Moreover, the small residual correlation
rescor (tarsus, back)at the bottom of the output suggests that there is little unmodeled dependence between tarsus length and back color. Although not necessary at this point, we have already calculated and stored its LOO information criterion
setting 1, which we will use for model comparisons. Next, let's take a look at some further predictive checks, which give us a first impression of the model's application.
pp_check(adjustment 1,resp = "hock")
pp_check(adjustment 1,resp = "to return")
This appears fairly stable, but we observe a slight unmodeled left skew in its distribution
hock. We will come back to this later. Next, we want to investigate how much variation in the response variables can be explained by our model, and we use a Bayesian generalization\(R^2\)coefficient.
Est.Est.Error Q2.5 Q97.5R2tarsus 0.4339755 0.02387684 0.3845041 0.4763270R2back 0.1980269 0.02823697 0.14072005
Clearly, there is a great deal of variation in the characteristics of both animals that we cannot explain, but apparently we can explain variation in tarsus length more than back color.
More complex multivariate models
Now suppose we just want to check for
hockbut not inside
to returnand vice versa for
Date of birth. Not that this makes much sense for the present example, but it allows us to show how to specify different types for different response variables. we can no longer use
mvbindsyntax and therefore we need to use a more comprehensive approach:
bf_tarsus<- friend(hock~sex+(1|Pi|adoption nest)+(1|q|barrier))bf_back<- friend(to return~Date of birth+(1|Pi|adoption nest)+(1|q|barrier))adjust 2<- brm(bf_tarsus+bf_back+ set_rescor(TRUE), data =BT data,chains = 2,cores = 2)
Note that we have literalsaddedthe two parts of the model through it
+operator, which in this case is equivalent to writing
mvbf(bf_tarsus, bf_back). Ver
help ("mvbrmsformula")for more details on this pension. Again, we summarize the model first.
adjust 2<- add_criterion(adjustment2,"toilet")summary(adaptation 2)
Family: MV (Gaussian, Gaussian) Links: mu = identity; sigma = identity mu = identity; sigma = identity Type: tarsus ~ sex + (1 | p | foster nest) + (1 | q | mother) back ~ date of birth + (1 | p | foster nest) + (1 | q | mother) Data: BTdata ( Number of observations: 828) Draws: 2 chains, each with iter = 2000; heating = 1000; skinny = 1; Total Bonds After Heat = 2000 Group Level Effects: ~barrier (number of levels: 106) Estimate Estimate. Error l-95% CI u-95% CI Rhat Bulk_ESSsd(tarsus_Intercept) 0.48 0.05 0.39 0.59 1.009 1.009 1.009 0.10 0.39 1.00 416cor(tarsus_Intercept,back_Intercept) -0.50 0.22 -0.90 -0.07 1.00 791 Tail_ESSsd(tarsus_Intercept) 1404sd(back_Intercept,back_Intercept,back_Intercepttar) number of levels: 10 4) EstimateSt.Error l-95% CI u-95% CI Rhat Bulk_ESSsd(tarsus_Intercept) ) 0.27 0.05 0.16 0.38 1.00 678sd(back_Intercept) 0.35 0.06 0.23 0.47 1.00 558Intercept)0.01 0.9 8 1.00 291 Tail_ESSsd (tarsus_Intercept) 1227sd(back_Intercept) 874cor(tar sus_Intercept, back_Intercept ) 681 Population Level Effects: Estimate Estimate Error l -95% CI u-95% CI Rhat Bulk_ESS Tail_ESStarsus_Intercept -0.41 0.07 -0.55 -0.28 1.00 2059 17801 .501 .501 . 0.00 2807 17 46tarsus_sexMale 0.77 0.06 0.66 0.88 1.00 4227 1517tarsus_sexUNK 0.23 0.13 -0.02 0.48 1.00 4517 1520back_hatchdate -0.08 0.05 -0. .19 0.02 1.00 3182 1470 Family Specific Parameters: Estimated-Error 5%CIg_ESSL ESS. ma_tarsus 0 .76 0.02 0. 72 0.79 1.00 2038 1659sigma_back 0.90 0.02 0.86 0.95 1.00 2406 1053 Residual correlations: Estimate est.error L -95% CI U -95% CI Rat Bulk_ess Tail_esscor (Tarsus, back) -0.05 0.04 -0.13 0.02 1.00 3349 1630 Mining Noted using (SUTS) sample. For each parameter, Bulk_ESS and Tail_ESS are effective measures of sample size, and Rhat is the potential downscaling factor in segregation chains (at convergence, Rhat = 1).
Let's find out how the model fit changed due to the exclusion of some effects from the original model:
Output of model 'fit1': Calculated from 2000 with 828 log-likelihood matrix Estimate SEelpd_loo -2126.0 33.6p_loo 175.6 7.4looic 4252.0 67.3-------MontepCarlo. . min. n_eff(-Inf, 0.5] (good) 803 97.0% 394 (0.5, 0.7] (ok) 23 2.8% 105 (0.7, 1] (bad) 2 0.2 % 26 (1, Inf) (very bad) 0 0.0%
See help ('pareto-k-diagnostic') for details. Output of 'fit2' model: Calculated from 2000 with 828 log-likelihood matrix Estimate SEelpd_loo -2123.2 33.7p_loo 173.8 7.5looic 4246.5 67.4------ The value of Monte Carlo di p. counting. Min. n_eff(-Inf, 0.5] (good) 809 97.7% 370 (0.5, 0.7] (ok) 17 2.1% 95 (0.7, 1] (bad) 2 0.2 % 28 (1, Inf) (very bad) 0 0.0% See help ('pareto-k-diagnostic') for details. Model comparisons: elpd_diff se_difffit2 0.0 0.0 fit1 -2.8 1.4
Obviously, there is no noticeable difference in the application of the model. So we don't really need to model
Date of birthfor both response variables, but it doesn't hurt to include them (so I probably would).
To give you a taste of its potentialbrmsmultivariate syntax, we change our model in many directions at once. Note its slight tilt to the left
hock, which we are now going to model using the
skew_normalfamily instead of
gaussianfamily. As we no longer have a normal multivariate (or t-student) model, estimation of residual correlations is no longer possible. We make this explicit by using
set_rescormode. In addition, we investigated whether the
Date of birthis indeed linear as we assumed earlier by fitting a non-linear spline to it
Date of birth. In addition, we model separate residual variances of
hockfor male and female chicks.
bf_tarsus<- friend(hock~sex+(1|Pi|adoption nest)+(1|q|barrier))+ with(Sigma~ 0 +sex)+ skew_normal()bf_back<- friend(to return~ small(hatch date)+(1|Pi|adoption nest)+(1|q|barrier))+ gaussian()ajuste3<- brm(bf_tarsus+bf_back+ set_rescor(FALSE), data =BT data,chains = 2,cores = 2, control = list(adapt_delta = 0,95))
Again, we summarize the model and consider some further predictive checks.
Family: MV(skew_normal, gaussian) Links: mu = identity; sigma = write; alpha = identity mu = identity; sigma = identity Type: tarsus ~ sex + (1 | p | foster nest) + (1 | q | mother) sigma ~ 0 + sex back ~ s(birthdate) + (1 | p | foster nest) + (1 | q|mother) Data: BTdata (Number of observations: 828) Plots: 2 strings, each with iter = 2000; heating = 1000; skinny = 1; Total Draws After Hot = 2000 Smooth Terms: Estimate Estimate. Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESSsds(back_shatchdate_1) 1.97 0.99 0.25 4.22 1.00 503 ~ Effect of 338 Levels ) Estimate Estimate Error l-95% CI u-95% CI Rhat Bulk_ESSsd(tarsus_Intercept) 0.48 0.05 0.39 0.58 1.00 832sd(back_Intercept) 0.24 0.07 0.07 0.19Intercept. ) -0.52 0.2 2 -0.93 -0.06 1.00 417 Tail_ESSsd(tarsus_Intercept ) 1395sd(back_Intercept ) 681cor(tarsus_Intercept,back_Intercept) 445~fosternest (Number of levels: 104) Estimate error l- 95% CI u-95% CI u-95% CI Rattark. 15 0.37 1. 00 535sd(back_Intercept) 0.31 0.06 0.20 0.42 1.00 512cor(tarsus_Intercept,back_Intercept) 0.65 0.22 0.13 0.98 1.00 255 Tail_Intercept(tail_intercept) 0,0,0,0,0,0,5) sus_Intercept,back_Intercept) 530Population Level Effects: Error Estimate l-95% CI u -95% CI Rhat Bulk_ESS Tail_ESStarsus_Intercept -0.41 0.07 -0.54 - 0.28 1.00 977 1420back_Intercept 0.00 0.05 -0.10 0.11 1.00 1475 0.54 -0.28 . 8 1.00 3099 1461tarsus_sexUNK 0.21 0.12 -0.02 0.45 1. 00 2394 1305sigma_tarsus_sexFemale -0.30 0.04 -0.38 -0.22 1.0 2537 1613sigma_tarsus_sexMale -0.25 0. 04 -0.33 -0.17 1.00 2200 1266sigma_tarsus_sexUNK -0.39 0. 13 -0.64 -0.12 1.013 -0.140 1831 6.14 6.59 1.00 1044 1029 Family parameters: Estimated error Estimated r l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESSsigma_back 0.90 0.02 0.85 0.95 1.00 2296 1301alpha_tarsus -1.22 0.43 -1.89 0.07 1.00 1626 682 sampling using NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective measures of sample size, and Rhat is the potential downscaling factor in segregation chains (at convergence, Rhat = 1).
We see that the residual (log) standard deviation of
hockit is slightly higher for chicks whose sex cannot be determined compared to male or female chicks. In addition, we see from the negatives
alpha(deformation) parameter of
hockthat the residuals are actually slightly skewed to the left. Finally, running
conditional_effect(just3,"hatch date",resp = "to return")
reveals a non-linear relationship of
Date of birthno
to returncolor, which seems to change in waves during birth dates.
There are many more modeling options for multivariate models that are not discussed in this vignette. Examples include autocorrelation structures, Gaussian processes, or explicit non-linear predictions (for example, see
βινιέτα ("brms_multililevel")). In fact, almost all the flexibility of univariate models is retained in multivariate models.
Hadfield JD, Nutall A, Osorio D, Owens IPF (2007). Testing the phenotypic gambit: phenotypic, genetic, and environmental correlates of color.Journal of Evolutionary Biology, 20(2), 549-557.
A multivariate model is a statistical tool that uses multiple variables to forecast outcomes. One example is a Monte Carlo simulation that presents a range of possible outcomes using a probability distribution. Black swan events rendering the model meaningless even if the data sets and variables being used are good.How do you create a multivariate regression model? ›
- Step 1: Select the features. First, you need to select that one feature that drives the multivariate regression. ...
- Step 2: Normalize the feature. ...
- Step 3: Select loss function and formulate a hypothesis. ...
- Step 4: Minimize the cost and loss function. ...
- Step 5: Test the hypothesis.
Ŷi = a + bXi is called the fitted (or predicted) value, and Yi Ŷi is called the residual. and the test for H0: β=0, is t = b / se(b) [p-value derived from t-distr.What is the difference between multivariate analysis and regression? ›
But when we say multiple regression, we mean only one dependent variable with a single distribution or variance. The predictor variables are more than one. To summarise multiple refers to more than one predictor variables but multivariate refers to more than one dependent variables.What are the 3 categories of multivariate analysis? ›
Multiple linear regression. Multiple logistic regression. Multivariate analysis of variance (MANOVA) Factor analysis.What is the most common multivariate analysis? ›
Multiple Regression Analysis
Multiple regression is the most commonly utilized multivariate technique. It examines the relationship between a single metric dependent variable and two or more metric independent variables.
The 3 most common types of multivariable regression are linear regression, logistic regression and Cox proportional hazards regression. A detailed understanding of multivariable regression is essential for correct interpretation of studies that utilize these statistical tools.What are multivariate regression techniques? ›
As the name implies, multivariate regression is a technique that estimates a single regression model with more than one outcome variable. When there is more than one predictor variable in a multivariate regression model, the model is a multivariate multiple regression.How to do multivariate analysis in Excel? ›
- Activate the Data Analysis ToolPak. After you open Excel, the first step is to ensure the Data Analysis ToolPak is active. ...
- Enter your basic data. The next step is to enter your basic data manually. ...
- Input your dependent data. ...
- Input your independent data. ...
- Execute your analysis.
Principal component analysis (PCA) and Factor analysis are two of the common techniques used to perform such a dimension reduction.
Conventionally, the minimum required sample size for almost all types of multivariable analysis is determined using a rule-of-thumb such as for MLR/ANCOVA (16–17), logistic regression (5–6) and exploratory factor analysis (18–20).What is a multivariate model? ›
Statistically speaking, multivariate analysis refers to statistical models that have 2 or more dependent or outcome variables,1 and multivariable analysis refers to statistical models in which there are multiple independent or response variables.What is the multivariate multiple regression model? ›
The multivariate multiple regression model is an extension of the standard multiple linear regression model. Multiple linear regression concerns predicting or explaining values of one response variable based on values of a collection of two or more predictor variables.How to interpret multivariate regression? ›
- The coefficients may or may not be statistically significant.
- The coefficients hold true on average.
- The coefficients imply association not causation.
- The coefficients control for other factors.
What are the disadvantages of multivariate analysis? Multivariate analysis sometimes requires more complex computations to arrive at an answer, and you must make sure you have enough data for all the variables you're analyzing.What are the main features of multivariate data analysis? ›
Most of multivariate analysis deals with estimation, confidence sets, and hypothesis testing for means, variances, covariances, correlation coefficients, and related, more complex population characteristics.What are the four multivariate test statistics? ›
In multivariate analysis there are four major test statistics, Wilks' Lambda, Pillai's Trace, the Hotelling-Lawley Trace, and Roy's Greatest Root.What are two applications of multivariate analysis? ›
Applications of multivariate analysis are found in almost all the disciplines which make up the bulk of policy-making, e.g. economics, healthcare, pharmaceutical industries, applied sciences, sociology, and so on.What are the two basic groups of multivariate techniques? ›
The basic types of multivariate techniques are metric methods and nonmetric methods.Which software is commonly used in multivariate analysis? ›
SIMCA® Multivariate Data Analysis software is the benchmark data analytics tool for scientists, engineers, researchers, product developers and others striving to gain information from large quantities of data. This data analysis software tool: Enables easy batch interpretation and analysis of large process data sets.
Regression analysis includes several variations, such as linear, multiple linear, and nonlinear. The most common models are simple linear and multiple linear.What is the difference between univariate model and multivariate model? ›
Univariate analysis looks at one variable, Bivariate analysis looks at two variables and their relationship. Multivariate analysis looks at more than two variables and their relationship.What is an example of a multivariate regression equation? ›
Multiple regression formulas analyze the relationship between dependent and multiple independent variables. For example, the equation Y represents the formula is equal to a plus bX1 plus cX2 plus dX3 plus E where Y is the dependent variable, and X1, X2, and X3 are independent variables.Is multivariate analysis hard? ›
Multivariate analysis can be complicated by the desire to include physics-based analysis to calculate the effects of variables for a hierarchical "system-of-systems". Often, studies that wish to use multivariate analysis are stalled by the dimensionality of the problem.How do you visualize multivariate data? ›
Another way of visualizing multivariate data for multiple attributes together is to use parallel coordinates. Basically, in this visualization as depicted above, points are represented as connected line segments. Each vertical line represents one data attribute.What is the formula for multivariate regression in Excel? ›
Using LINEST() Function in Excel for Multivariate Regression
The equation for the line is y = mx + b or y = m1x1 + m2x2 + ... + b (if there are multiple ranges of x values) where the dependent y value is a function of the independent x values.
Multivariate methods are not the same as multivariable methods. Multivariate methods have more than one dependent variable or place variables on an equal footing. Multivariable methods have one dependent variable and more than one independent variables or covariates.Is two way ANOVA a multivariate technique? ›
The two-way multivariate analysis of variance (two-way MANOVA) is often considered as an extension of the two-way ANOVA for situations where there is two or more dependent variables.Is logistic regression a multivariate analysis? ›
This is done using “multivariable logistic regression” – a technique that allows us to study the simultaneous effect of multiple factors on a dichotomous outcome.What is the minimum number of variables required for multivariate analysis? ›
For linear regressions, i. e. multivariable analyses for which the outcome variable is numerical, it is necessary to have at least 10 observations per covariate. A small refinement, when the covariate is categorical with N classes, it counts as N-1 variables.
Many difficulties tend to arise when there are more than five independent variables in a multiple regression equation. One of the most frequent is the problem that two or more of the independent variables are highly correlated to one another. This is called multicollinearity.How many dependent variables are in multivariate analysis? ›
Multivariate Analysis of Variance and Covariance
The hypothesis concerns a comparison of vectors of group means. A MANOVA has one or more factors (each with two or more levels) and two or more dependent variables.
The most important advantage of Multivariate regression is it helps us to understand the relationships among variables present in the dataset. This will further help in understanding the correlation between dependent and independent variables. Multivariate linear regression is a widely used machine learning algorithm.What are the assumptions of multivariate regression models? ›
Five main assumptions underlying multiple regression models must be satisfied: (1) linearity, (2) homoskedasticity, (3) independence of errors, (4) normality, and (5) independence of independent variables. Diagnostic plots can help detect whether these assumptions are satisfied.How do you choose variables for multivariate regression? ›
- Variables that are already proven in the literature to be related to the outcome.
- Variables that can either be considered the cause of the exposure, the outcome, or both.
- Interaction terms of variables that have large main effects.
R-Squared (R² or the coefficient of determination) is a statistical measure in a regression model that determines the proportion of variance in the dependent variable that can be explained by the independent variable. In other words, r-squared shows how well the data fit the regression model (the goodness of fit).What is a real life example of multivariate data? ›
Examples of multivariate regression
Example 2. A doctor has collected data on cholesterol, blood pressure, and weight. She also collected data on the eating habits of the subjects (e.g., how many ounces of red meat, fish, dairy products, and chocolate consumed per week).
Multivariate refers to multiple dependent variables that result in one outcome. This means that a majority of our real-world problems are multivariate. For example, based on the season, we cannot predict the weather of any given year. Several factors play an important role in predicting the same.What is a practical example of multivariate analysis? ›
An example would be to determine the factors that predict the selling price or value of an apartment. Multiple linear correlation: Allows for the determination of the strength of the strength of the linear relationship between Y and a set of X variables.Why do we use multivariate models? ›
Multivariable regression models are used to establish the relationship between a dependent variable (i.e. an outcome of interest) and more than 1 independent variable. Multivariable regression can be used for a variety of different purposes in research studies.
Multivariate analysis is based in observation and analysis of more than one statistical outcome variable at a time. In design and analysis, the technique is used to perform trade studies across multiple dimensions while taking into account the effects of all variables on the responses of interest.How do multivariate models work? ›
Multivariate models are used to decompose the covariance between two or more traits into genetic and environmental sources and to estimate the amount of overlapping genetic and environmental influences on the traits (genetic and environmental correlations).What is multivariate in data analysis? ›
Multivariate analysis is the study of multiple variables in a set of data. Variables are factors you compare to the control or unchanging component of the experiment. Variables help you compare your findings with the control of the experiment to identify any changes that might occur or trends that may develop.What is an example of a multivariate hypothesis test? ›
Common examples of multivariate tests include: Testing text and visual elements on a webpage together. Testing the text and color of a CTA button together. Testing the number of form fields and CTA text together.What is an example of multivariate random variables? ›
Multivariate Discrete Random Variables
In other words, multivariate random variables are vectors of random variables. For instance, a bivariate random variable X can be a vector with two components X1 and X2 with the corresponding realizations being x1 and x2 , respectively.