# Function to conduct mediation analysis with micro effects on macro structure (MEMS). `mediate_MEMS` implements parametric and nonparametric routines to compare `MEMS` estimate between models. Largely a wrapper for `compare_MEMS`, the sole novel functionality of `mediate_MEMS` is provided when a user specifies the `model_comparison` argument to be `TRUE`. When `model_comparison` is set to `TRUE`, `mediate_MEMS` compares the direct, indirect, and total `MEMS` estimates to those obtained from `partial_model2` and `full_model2`. This can be used as a sensitivity test to researchers' choice of model. It can also provide the basis for testing differences in direct, total, and indirect effect sizes within the same model by setting `partial_model2` to equal `partial_model` and `full_model2` to equal `full_model` and provide a second explanatory micro process to the `micro_process2` argument. In these cases, the difference between "models" captures the differences effect size of the direct, indirect, and total `MEMS` estimates for two distinct explanatory micro processes. See Wertsching and Duxbury (2025) for details. The difference in MEMS is the change in MEMS after one or more micro-processes are included into a model or, in the case of sensitivity tests, when the functional form is changed. Let $MEMS_p$ represent the MEMS obtained from a model that omits one or more intervening variables and $MEMS_f$ be the MMES obtained from a model that includes the intervening variable(s). The change in MEMS is given [REMOVE_ME]$$ \Delta MEMS=MEMS_p-MEMS_f [REMOVE_ME_2]$$. $MEMS_p$ and $MEMS_f$ may also be have the same specification but use distinct functional forms or other modeling decisions in the case of sensitivity tests. Tuning parameters can be assigned to toggle the strength of $\theta$ in model-implied estimates of $MEMS$. `MEMS` currently accepts `glm`, `glmer`, `ergm`, `btergm`, `sienaFit`, `rem.dyad`, and `netlogit` objects and implements both parametric and nonparametric estimation. Pooled estimation for multiple network models is also implemented for `ergm` and `sienaFit` objects. ## Description `mediate_MEMS` implements parametric and nonparametric routines to compare `MEMS` estimate between models. Largely a wrapper for `compare_MEMS`, the sole novel functionality of `mediate_MEMS` is provided when a user specifies the `model_comparison` argument to be `TRUE`. When `model_comparison` is set to `TRUE`, `mediate_MEMS` compares the direct, indirect, and total `MEMS` estimates to those obtained from `partial_model2` and `full_model2`. This can be used as a sensitivity test to researchers' choice of model. It can also provide the basis for testing differences in direct, total, and indirect effect sizes within the same model by setting `partial_model2` to equal `partial_model` and `full_model2` to equal `full_model` and provide a second explanatory micro process to the `micro_process2` argument. In these cases, the difference between "models" captures the differences effect size of the direct, indirect, and total `MEMS` estimates for two distinct explanatory micro processes. See Wertsching and Duxbury (2025) for details. The difference in MEMS is the change in MEMS after one or more micro-processes are included into a model or, in the case of sensitivity tests, when the functional form is changed. Let $MEMS_p$ represent the MEMS obtained from a model that omits one or more intervening variables and $MEMS_f$ be the MMES obtained from a model that includes the intervening variable(s). The change in MEMS is given $$ \Delta MEMS=MEMS_p-MEMS_f $$ . $MEMS_p$ and $MEMS_f$ may also be have the same specification but use distinct functional forms or other modeling decisions in the case of sensitivity tests. Tuning parameters can be assigned to toggle the strength of $\theta$ in model-implied estimates of $MEMS$. `MEMS` currently accepts `glm`, `glmer`, `ergm`, `btergm`, `sienaFit`, `rem.dyad`, and `netlogit` objects and implements both parametric and nonparametric estimation. Pooled estimation for multiple network models is also implemented for `ergm` and `sienaFit` objects. ```r mediate_MEMS(partial_model, full_model, micro_process, macro_function, model_comparison=FALSE, partial_model2=NULL, full_model2=NULL, micro_process2=NULL, macro_function2=NULL, object_type=NULL, interval=c(0,1), nsim=500, algorithm="parametric", silent=FALSE, full_output=FALSE, sensitivity_ev=TRUE, SAOM_data=NULL, SAOM_var=NULL, time_interval=NULL, covar_list=NULL, edgelist=NULL, net_logit_y=NULL, net_logit_x=NULL, group_id=NULL, node_numbers=NULL, mediator=NULL, link_id=NULL, controls=NULL, control_functions=NULL) ``` ## Arguments - `partial_model`: the micro-model excluding one or more intervening or confounding variables of interest. May also be a fully specified model with a distinct functional form in the case of sensitivity tests. Currently accepts `glm`, `glmer`, `ergm`, `btergm`, `sienaFit`, `rem.dyad`, and `netlogit` objects. Pooled estimation for multiple network models is also implemented for `ergm` and `sienaFit` objects. To implement pooled estimation, `model` should be provided as a list of `ergm` or `sienaFit` objects. - `full_model`: the micro-model including one or more intervening or confounding variables of interest. May also be a fully specified model with a distinct functional form in the case of sensitivity tests. Currently accepts `glm`, `glmer`, `ergm`, `btergm`, `sienaFit`, `rem.dyad`, and `netlogit` objects. Pooled estimation for multiple network models is also implemented for `ergm` and `sienaFit` objects. To implement pooled estimation, `model` should be provided as a list of `ergm` or `sienaFit` objects. - `micro_process`: a character string containing the name of the micro process of interest. The character string should exactly match coefficient names in `model` output. - `macro_function`: a `function` that calculates the macro statistic of interest. Currently accepts user defined functions as well as functions inherent in the `igraph` and `statnet` packages for . - `model_comparison`: returns sensitivity tests evaluating robustness of partial, full, and indirect MEMS estimates to distinct model choices when set to `TRUE`. Default is `FALSE` - `partial_model2`: a second model identitical to the specification of `partial_model` that uses a distinct functional form. Differences in MEMS estimates from `partial_model` are compared to MEMS estimates from `partial_model2`. Required if `model_comparison=TRUE` and ignored otherwise. - `full_model2`: a second model identitical to the specification of `full_model` that uses a distinct functional form. Differences in MEMS estimates from `full_model` are compared to MEMS estimates from `full_model2`. Required if `model_comparison=TRUE` and ignored otherwise. - `micro_process2`: the character string identifying the micro_process to compare in sensitivity analysis. Required if `model_comparison=TRUE` and ignored otherwise. - `macro_function2`: an optional function that calculates the macro statistic of interest. When provided, `macro_function2` will be used in place of `macro_function` when calculating the `MEMS` provided by the `partial_model2` and `full_model2` arguments. This is intended for use in comparisons between distinct models (e.g., SAOM and ERGM) when a user provided function must be written differently for each model's output. Defaults to `NULL`. - `object_type`: A character string that tells netmediate the type of object to apply the `macro_function` to. Currently accepts `igraph` and `network` objects. If left `NULL`, `network` objects are assumed. Can be over-ridden to use other object types with a user-function by defining a function that accepts either a `network` or `igraph` object and returns a numeric value or vector of numeric values (see examples). - `interval`: The value of tuning parameters to assign to $\theta$. Should be provided as a vector of numeric values with 2 entries. - `nsim`: The number of simulations or bootstrap samples to use during estimation. - `algorithm`: The estimation algorithm to be used. Currently accepts `"parametric"` and `"nonparametric"`. If `"parametric"`, estimation is obtained with Monte Carlo sampling. If `"nonparametric"`, estimation uses bootstrap resampling. - `silent`: logical parameter. Whether to provide updates on the progress of the simulation or not. - `full_output`: logical parameter. If set to `TRUE`, `mediate_MEMS` will return all sampled statistics and complete results for $MEMS_p$ and $MEMS_f$. - `sensitivity_ev`: optional parameter. If set to `TRUE`, will return E-values for direct, total, and indirect MEMS estimates. - `SAOM_data`: required when the model is a `sienaFit` object; ignored otherwise. If a `sienaFit` object is provided, `SAOM_data` should be the `siena` object that contains the data for SAOM estimation. If using pooled estimation on multiple `sienaFit` objects (i.e., providing a list of `sienaFit` objects), then `SAOM_data` should be provided as an ordered list with each entry containing the `siena` object corresponding to list of `sienaFit` objects. - `SAOM_var`: optional parameter when the model is a `sienaFit` object. `SAOM_var` is a list of of the `varCovar` and `varDyadCovar` objects used to assign time varying node and dyad covariates when calling `sienaDataCreate`. If provided, `netmediate` assigns the varying node covariates and dyad covariates to each simulated network. This parameter is required when `macro_function` computes a statistic that varies as a function of time varying node or dyad covariates (i.e., network segregation, assorativity). Time invariant characteristics (`coCovar` and `coDyadCovar`) are handled internally by `MEMS` and should not be provided. When providing a list of `sienaFit` objects for pooled estimation, `SAOM_var` should be provided as a list of lists, where each entry in the list contains a list of `varCovar` and `varDyadCovar` objects associated with corresponding `sienaFit` object. - `time_interval`: an optional parameter to be used with `rem.dyad` objects. May be provided as a numeric vector or the character string `"aggregate"`. If a numeric vector is provided unique network snapshots at each interval. For example, `time_interval=c(0,2,3)` would induce two networks, one for the 0 - 2 time period and one for the 2 - 3 time period. If specified as `"aggregate"`, the MEMS is calculated by creating an aggregated cross-sectional representation of the entire event sequence. If left `NULL`, defaults to |`"aggregate"`. - `covar_list`: an optional list of sender/receiver covariates used in `rem.dyad` estimation. Only required for `rem.dyad` objects when covariates are included. The list format should correspond to the format required by `rem.dyad` - `edgelist`: an optional three column edgelist providing the sender, receiver, and time of event occurrence when using rem.`rem.dyad`. Only required when `time_interval` is set to `NULL` or `"aggregate"`. Ignored for other types of models. - `net_logit_y`: the dependent variable for `netlogit` objects. Should be provided as a vector. Only required when model is a `netlogit` object. - `net_logit_x`: the matrix of independent variables for `netlogit` type objects. Only required when model is a `netlogit` object. - `group_id`: optional vector of group identifiers to use when estimating a `glm` or `glmer` on grouped data (i.e., multiple time periods, multiple networks). When specified, `MEMS` will induce unique networks for each grouping factor. If left unspecified, all groups/time periods are pooled. If using `glmer`, the grouping factor does not have to be provided as part of the model or used as a random effect. - `node_numbers`: a numeric vector containing the number of nodes in each group_id when using `glm` or `glmer`. If estimating MEMS aggregated over all networks (i.e., `group_id=NULL`), this shoud be the total number of nodes in all networks. Required when using `glm` or `glmer`, ignored otherwise. - `mediator`: a character string detailing the mediator of interest. Intended for internal use with the `AMME` function; not intended for end users. - `link_id`: a vector or list of vectors corresponding to unique identifiers. Intended for internal use with the `AMME` function; not intended for end users. - `controls`: a vector of character strings listing the controls to be calculated when using `AMME`. Intended for internal use with the `AMME` function; not intended for end users. - `control_functions`: a list of functions to calculate the macro control variables provided in controls. Intended for internal use with the `AMME` function; not intended for end users. ## Details Compares MEMS estimates between two models. If one or more confounding or intervening variables are excluded or included between models, the change in MEMS can be interpreted as the portion of the MEMS explained by one or more confounding or intervening variable. If two models are provided with the same specification but a distinct functional form, the change in MEMS is a sensitivty test of how much the MEMS estimate changes because of a model decision. This can be useful, for example, when comparing TERGM and SAOM estimates as each models make distinct assumptions about sources of network change and the temporal ordering of tie changes. If a single pair of models are compared, `mediate_MEMS` results will be identical to `compare_MEMS` results. However, if `model_comparison` is set to `TRUE` and a second pair of models are provided using a different set of control variables or a distinct functional form, `mediate_MEMS` will also return sensitivity tests evaluating whether the partial, full, and indirect MEMS estimates are significantly different between the two sets of models. This functionality can also be used to formally test differences in partial, full, and indirect effect sizes by setting `partial_model2` to be the same as `partial model` and `full_model2` to be the same as `full_model` and setting `micro_process2` to be a different variable than `micro_process`. `mediate_MEMS` functionality inherits directly from the `compare_MEMS` and `MEMS` commands. See the `compare_MEMS` and `MEMS` pages for more details. ## Returns If `full_output=FALSE` and `model_comparison=FALSE`, then a table is returned with the change MEMS, its standard error, confidence interval, and p-value, and the same results for the partial and complete MEMS. If `full_output=TRUE` and `model_comparison=FALSE`, then a list is returned with the following three elements. - **diff_MEMS_results**: is the table of summary output containing the MEMS, its standard error, confidence interval, and p-value, and a list of the simulated values of the change in MEMS. - **p_MEMS_results**: contains the summary statistics for the partial MEMS along with all simulated statistics. - **f_MEMS_results**: contains the summary statistics for the full MEMS along with all simulated statistics. If `full_output=FALSE` and `model_comparison=TRUE`, then a list is returned with the following elements. - **model_set1**: is the table of summary output testing the partial, full, and indirect MEMS using `partial_model` and `full_model`. - **model_set2**: is the table of summary output testing the partial, full, and indirect MEMS using `partial_model2` and `full_model2` - **model_comparison**: is the table of output testing the difference in estimates between `model_set1` and `model_set2` If `full_output=TRUE` and `model_comparison=TRUE`, then a list is returned with the following three elements. - **summary_results**: is the list of summary output provided as described above when `full_output=TRUE`. - **sample_results**: is a list containing the simulation draws used to calculate point estimates, variance estimates, and sensitivity tests for the partial MEMS, full MEMS, and indirect MEMS. ## References Duxbury, Scott W. 2024. "Micro Effects on Macro Structure in Social Networks." Sociological Methodology. Wertsching, Jenna, and Scott W. Duxbury. Working paper. "Micro Effects on Macro Structure: Identification, Comparison between Nested Models, and Sensitivity Tests for Functional Form." Duxbury, Scott W., and Xin Zhao. Working paper. "Sensitivity Tests for Micro-Macro Network Analysis." ## Author(s) Duxbury, Scott W. Associate Professor, University of North Carolina--Chapel Hill, Department of Sociology. ## See Also `AMME` `MEMS` `ergm.mma` `mediate` `compare_MEMS` ## Examples ```r ## Not run: library(statnet) library(igraph) data("faux.mesa.high") #how much of the effect of racial homophily on transitivity #is explained by triadic closure effects? model<-ergm(faux.mesa.high~edges+nodecov("Grade")+nodefactor("Race")+ nodefactor("Sex")+nodematch("Race")+nodematch("Sex")+absdiff("Grade")) model2<-ergm(faux.mesa.high~edges+nodecov("Grade")+nodefactor("Race")+ nodefactor("Sex")+nodematch("Race")+nodematch("Sex")+absdiff("Grade")+ gwesp(.5,fixed=TRUE)) #compare results from compare_MEMS and mediate_MEMS compare_MEMS(partial_model=model, full_model=model2, micro_process="nodematch.Race", macro_function=transitivity, object_type = "igraph", silent=FALSE, algorithm="parametric") mediate_MEMS(partial_model=model, full_model=model2, micro_process="nodematch.Race", macro_function=transitivity, object_type = "igraph", silent=FALSE, algorithm="parametric") ###test sensitivity to MPLE versus MCMC MLE estimation modela<-ergmMPLE(faux.mesa.high~edges+nodecov("Grade")+nodefactor("Race")+ nodefactor("Sex")+nodematch("Race")+nodematch("Sex")+absdiff("Grade"), output="fit") model2a<-ergmMPLE(faux.mesa.high~edges+nodecov("Grade")+nodefactor("Race")+ nodefactor("Sex")+nodematch("Race")+nodematch("Sex")+absdiff("Grade")+ gwesp(.5,fixed=TRUE), output="fit") mediate_MEMS(partial_model=model, full_model=model2, micro_process="nodematch.Race", model_comparison = TRUE, partial_model2=modela, full_model2=model2a, micro_process2="nodematch.Race", macro_function=transitivity, object_type = "igraph", silent=FALSE, algorithm="parametric") ##compare direct, total, and indirect effect sizes mediate_MEMS(partial_model=model, full_model=model2, micro_process="nodematch.Race", model_comparison = TRUE, partial_model2=model, full_model2=model2, micro_process2="absdiff.Grade", macro_function=transitivity, object_type = "igraph", silent=FALSE, algorithm="parametric") ##################################################### # More complicated sensitivty test using macro function 2 ##################################################### #are the direct, total, and indirect MEMS of #network selection on similar smoking behavior on #students' shared smoking robust to distinct #model choices? ###Compare between SAOM and TERGM treating behavioral #(smoking) autocorrelation as outcome, smoking homophily #as treatment, and triadic closure as mediator ######################################## # Co-evolution SAOM ####################################### library(RSiena) network_array<-list(s501,s502,s503) smoking<-as.data.frame(s50s) for(i in 1:ncol(smoking)){ smoking[,i][smoking[,i]>1]<-2 } alcohol<-as.data.frame(s50a) ##we'll use alcohol consumption as a covariate as well ##create "sienaDependent" object, TLSnet<-sienaDependent(array(c(network_array[[1]], network_array[[2]], network_array[[3]]), dim=c(50,50,3))) TLSbeh<-sienaDependent(as.matrix(smoking),type="behavior") #set covariates Alcohol<-varCovar(as.matrix(alcohol)) ###create dataset, but specify network AND behavior SAOM.Data<-sienaDataCreate(Network=TLSnet, Behavior=TLSbeh, Alcohol) ###Create the effects object SAOM.terms<-getEffects(SAOM.Data) ###We'll start by specifying the NETWORK function SAOM.terms<-includeEffects(SAOM.terms,egoX,altX,absDiffX,interaction1="Alcohol") SAOM.terms<-includeEffects(SAOM.terms,egoX,altX,sameX,interaction1="Behavior") ###Now let's specify the BEHAVIOR function SAOM.terms<-includeEffects(SAOM.terms,effFrom,name="Behavior", interaction1="Alcohol") SAOM.terms<-includeEffects(SAOM.terms,totSim,name="Behavior", interaction1="Network") SAOM.terms<-includeEffects(SAOM.terms,isolate, name="Behavior",interaction1="Network") #estimate the model WITHOUT transitive ties create.model<-sienaAlgorithmCreate(projname="Co-evolution_output", seed=21093, nsub=4, n3=1000) TLSmodel_notrans<-siena07(create.model, data=SAOM.Data, effects=SAOM.terms, verbose=TRUE, returnDeps=TRUE) #include transTies SAOM.terms<-includeEffects(SAOM.terms,transTies,inPop) TLSmodel<-siena07(create.model, data=SAOM.Data, effects=SAOM.terms, verbose=TRUE, returnDeps=TRUE) #now fit the TERGM library(statnet) net_list<-list(as.network(s501),as.network(s502),as.network(s503)) net_list[[1]]<-network::set.vertex.attribute(net_list[[1]],"smoking",s50s[,1]) net_list[[2]]<-network::set.vertex.attribute(net_list[[2]],"smoking",s50s[,2]) net_list[[3]]<-network::set.vertex.attribute(net_list[[3]],"smoking",s50s[,3]) net_list[[1]]<-network::set.vertex.attribute(net_list[[1]],"alcohol",s50a[,1]) net_list[[2]]<-network::set.vertex.attribute(net_list[[2]],"alcohol",s50a[,2]) net_list[[3]]<-network::set.vertex.attribute(net_list[[3]],"alcohol",s50a[,3]) TERGM_1_nogwesp<-tergm(net_list~Form( ~edges+ mutual+ gwidegree(.5,fixed=TRUE)+ nodeicov("smoking")+ nodeocov("smoking")+ nodematch("smoking")+ nodeicov("alcohol")+ nodeocov("alcohol")+ absdiff("alcohol")), estimate="CMLE" ) TERGM_1<-tergm(net_list~Form( ~edges+ mutual+ gwesp(.5,fixed=TRUE)+ gwidegree(.5,fixed=TRUE)+ nodeicov("smoking")+ nodeocov("smoking")+ nodematch("smoking")+ nodeicov("alcohol")+ nodeocov("alcohol")+ absdiff("alcohol")), estimate="CMLE" ) #create network autocorrelation function for TERGM Moran_tergm<-function(x){ y<-network::get.vertex.attribute(x,"smoking") return(nacf(x,y,type="moran",lag=1)[2]) } #test difference in TERGM and SAOM direct, total, and indirect #MEMS estimates mediate_MEMS(partial_model=TLSmodel_notrans, full_model=TLSmodel, micro_process="same Behavior", macro_function =Moran_dv, model_comparison = TRUE, partial_model2=TERGM_1_nogwesp, full_model2=TERGM_1, micro_process2="Form(1)~nodematch.smoking", macro_function2=Moran_tergm, object_type = "network", SAOM_data = SAOM.Data, silent=FALSE, nsim=100) ## End(Not run) ```

mediate_MEMS function