# Compute the similarity test. This function compute the nonparametric test for spatial independence using symbolic analysis for categorical/qualitative spatial process. ```r similarity.test(formula = NULL, data = NULL, fx = NULL, listw = listw, alternative = "two.sided", distr = "asymptotic", nsim = NULL, control = list()) ``` ## Arguments - `formula`: a symbolic description of the factor (optional). - `data`: an (optional) data frame or a sf object containing the variable to testing for. - `fx`: a factor (optional). - `listw`: a listw object - `alternative`: a character string specifying the type of cluster, must be one of "High" (default), "Both" or "Low". - `distr`: A string. Distribution of the test "asymptotic" (default) or "bootstrap" - `nsim`: Number of permutations. - `control`: List of additional control arguments. ## Returns A object of the **htest** ||| |:--|:--| |`data.name`|a character string giving the names of the data.| |`statistic`|Value of the similarity test| |`N`|total number of observations.| |`Zmlc`|Elements in the Most Likelihood Cluster.| |`alternative`|a character string describing the alternative hypothesis.| |`p.value`|p-value of the similarity test| |`similiarity.mc`|values of the similarity test in each permutation.| ## Details This testing approach for spatial independence that extends some of the properties of the join count statistic. The premise of the tests is similar to the join count statistic in that they use the concept of similarity between neighbouring spatial entities (Dacey 1968; Cliff and Ord 1973, 1981). However, it differs by taking into consideration the number of joins belonging locally to each spatial unit, rather than the total number of joins in the entire spatial system. The approach proposed here is applicable to spatial and network contiguity structures, and the number of neighbors belonging to observations need not be constant. We define an equivalence relation $\sim$ in the set of locations S. An equivalence relation satisfies the following properties: Reflexive: $s_i \sim s_i$ for all $s_i \in S$, Symmetric: If $s_i \sim s_j$, then $s_j \sim s_i$ for all $s_i,\ s_j \in S$ and Transitive: If $s_i \sim s_j$ and $s_j \sim s_k$, then $s_i \sim s_k$ for all $s_i, \ s_j, \ s_k \in S$ We call the relation $\sim$ a similarity relation. Then, the null hypothesis that we are interested in is $$ H_0: \{X_s\}_{s \in S} \ \ is\ \ iid $$ Assume that, under the null hypothesis, $P(s_i \sim s_{ji}) = p_i$ for all $s_{ji} \in N_{s_i}$. Define $$ I_{ij}=1 \ \ if \ \ s_i \sim s_{ji} \ \ ; 0 \ \ otherwise $$ Then, for a fixed degree d and for all location si with degree d, the variable d $$ \Lambda_{(d,i)}=\sum_{j=1}^d I_{ij} $$ gives the number of nearest neighbours to si that are similar to si. Therefore, under the null hypothesis, $H_0$, $\Lambda(d,i)$ follows a binomial distribution $B(d, p_i)$. ## Control arguments ||| |:--|:--| |`seedinit`|Numerical value for the seed (only for boot version). Default value seedinit=123| ## Examples ```r # Case 1: N <- 100 cx <- runif(N) cy <- runif(N) listw <- spdep::knearneigh(cbind(cx,cy), k = 3) p <- c(1/4,1/4,1/4,1/4) rho <- 0.5 fx <- dgp.spq(p = p, listw = listw, rho = rho) W <- (spdep::nb2mat(spdep::knn2nb(listw)) >0)*1 similarity <- similarity.test(fx = fx, data = FastFood.sf, listw = listw) print(similarity) # Case 2: test with formula, a sf object (points) and knn data("FastFood.sf") coor <- sf::st_coordinates(sf::st_centroid(FastFood.sf)) listw <- spdep::knearneigh(coor, k = 4) formula <- ~ Type similarity <- similarity.test(formula = formula, data = FastFood.sf, listw = listw) print(similarity) # Case 3: data(provinces_spain) listw <- spdep::poly2nb(as(provinces_spain,"Spatial"), queen = FALSE) provinces_spain$Mal2Fml <- factor(provinces_spain$Mal2Fml > 100) levels(provinces_spain$Mal2Fml) = c("men","woman") formula <- ~ Mal2Fml similarity <- similarity.test(formula = formula, data = provinces_spain, listw = listw) print(similarity) ``` ## References * Farber, S., Marin, M. R., & Paez, A. (2015). Testing for spatial independence using similarity relations. **Geographical Analysis**. 47(2), 97-120. ## See Also `sp.runs.test`, `dgp.spq`, `Q.test`, , `scan.test` ## Author(s) ||| |:--|:--| |Fernando López|fernando.lopez@upct.es| |Román Mínguez|roman.minguez@uclm.es| |Antonio Páez|paezha@gmail.com| |Manuel Ruiz|manuel.ruiz@upct.es|

similarity.test function