On a Facebook methods group, there was a question about testing hypotheses in networks. In the comments, it was suggested that **BGGM** could be used to test the hypothesis. And it turns out that **BGGM** really shines for testing expectations (see for example Rodriguez et al. 2020).

In this vignette, I demonstrate three ways to go about testing the same hypothesis, which is essentially testing for a difference in the **sum** of partial correlations between groups.

```
# need the developmental version
if (!requireNamespace("remotes")) {
install.packages("remotes")
}
# install from github
remotes::install_github("donaldRwilliams/BGGM")
library(BGGM)
```

For demonstrative purposes, I use the `bfi`

data and test the hypotheses in males and females

```
# data
Y <- bfi
# males
Y_males <- subset(Y, gender == 1, select = -c(education, gender))[,1:5]
# females
Y_females <- subset(Y, gender == 2, select = -c(education, gender))[,1:5]
```

The first approach is rather straightforward, with the caveat that the method needs to be implemented by the user. Note that I could certainly implement this in **BGGM**, assuming there is enough interest. Please make a feature request here.

The hypothesis was that a sum of relations was larger in one group, for example,

\[
\begin{align}
\mathcal{H}_0: (\rho^{male}_{A1--A2}\; + \; \rho^{male}_{A1--A3}) = (\rho^{female}_{A1--A2}\; + \; \rho^{female}_{A1--A3}) \\
\mathcal{H}_1: (\rho^{male}_{A1--A2}\; + \; \rho^{male}_{A1--A3}) > (\rho^{female}_{A1--A2}\; + \; \rho^{female}_{A1--A3})
\end{align}
\] Note that the hypothesis is related to the sum of relations, which is readily tested in **BGGM**.

The first step is to estimate the model for each group

```
# fit female
fit_female <- estimate(Y_females, seed = 2)
# fit males
fit_male <- estimate(Y_males, seed = 1)
```

For an example, I used the default which is to assume the data is Gaussian. This can be changed with `type =`

either `binary`

, `ordinal`

, or `mixed`

.

The next step is to extract the posterior samples for each relation

```
post_male <- posterior_samples(fit_male)[,c("A1--A2", "A1--A3")]
post_female <- posterior_samples(fit_female)[,c("A1--A2", "A1--A3")]
```

Note that the column names reflect the upper-triangular elements of the partial correlation matrix. Hence, the first name (e.g.,`A1`

) must be located before the second name (e.g., `A2`

) in the data matrix. This can be understood in reference to the column numbers: `1--2`

is correct whereas `2--1`

will result in an error.

The next step is to sum the relations and compute the difference

```
# sum males
sum_male <- rowSums(post_male)
# sum females
sum_female <- rowSums(post_female)
# difference
diff <- sum_male - sum_female
```

which can then be plotted

```
# three column
par(mfrow=c(1,3))
# male sum
hist(sum_male)
# female sum
hist(sum_female)
# difference
hist(diff)
```

Next compute the posterior probability the sum is larger in males than females

```
# posterior prob
mean(sum_male > sum_female)
#> 0.737
```

and then the credible interval for the difference

```
quantile(diff, probs = c(0.025, 0.975))
#> 2.5% 97.5%
#> -0.06498586 0.12481253
```

The next approach is based on a posterior predictive check. The hypothesis is essentially the same as above, but for the predictive distribution, that is,

\[ \begin{align} \mathcal{H}_0: (\rho^{male^{yrep}}_{A1--A2}\; + \; \rho^{male^{yrep}}_{A1--A3}) = (\rho^{female^{yrep}}_{A1--A2}\; + \; \rho^{female^{yrep}}_{A1--A3}) \\ \mathcal{H}_1: (\rho^{male^{yrep}}_{A1--A2}\; + \; \rho^{male^{yrep}}_{A1--A3}) > (\rho^{female^{yrep}}_{A1--A2}\; + \; \rho^{female^{yrep}}_{A1--A3}) \end{align} \] where the only difference is \(yrep\). See more details here.

The first step is to define a function to compute the difference in sums

```
# colnames
cn <- colnames(Y_males)
# function
f <- function(Yg1, Yg2){
# data
Yg1 <- na.omit(Yg1)
Yg2 <- na.omit(Yg2)
# estimate partials
fit1 <- pcor_mat(estimate(Yg1, analytic = TRUE))
fit2 <- pcor_mat(estimate(Yg2, analytic = TRUE))
# names (not needed)
colnames(fit1) <- cn
rownames(fit1) <- cn
colnames(fit2) <- cn
rownames(fit2) <- cn
# take sum
sum1 <- fit1["A1", "A2"] + fit1["A1", "A3"]
sum2 <- fit2["A1", "A2"] + fit2["A1", "A3"]
# difference
sum1 - sum2
}
```

Note that the function takes two data matrices and then returns a single value. Also, the default in **BGGM** does not require a custom function (only needs the data from each group).

The next step is to compute the observed difference and then perform the check.

```
# observed
obs <- f(Y_males, Y_females)
# check
ppc <- ggm_compare_ppc(Y_males, Y_females,
iter = 250,
FUN = f,
custom_obs = obs)
# print
ppc
#> BGGM: Bayesian Gaussian Graphical Models
#> ---
#> Test: Global Predictive Check
#> Posterior Samples: 250
#> Group 1: 896
#> Group 2: 1813
#> Nodes: 5
#> Relations: 10
#> ---
#> Call:
#> ggm_compare_ppc(Y_males, Y_females, iter = 250, FUN = f, custom_obs = obs)
#> ---
#> Custom:
#>
#> contrast custom.obs p.value
#> Yg1 vs Yg2 0.029 0.264
#> ---
```

Note this requires the user to determine \(\alpha\).

The check can also be plotted

`plot(ppc)`

where the red is the critical region.

The above approaches cannot provide evidence that the sum is equal. In other words, just because there was not a difference, this does not provide evidence for equality. The Bayes factor methods allow for formally assessing the equality model, that is,

\[ \begin{align} \mathcal{H}_1&: (\rho^{male}_{A1--A2}\; + \; \rho^{male}_{A1--A3}) > (\rho^{female}_{A1--A2}\; + \; \rho^{female}_{A1--A3}) \\ \mathcal{H}_2&: (\rho^{male}_{A1--A2}\; + \; \rho^{male}_{A1--A3}) = (\rho^{female}_{A1--A2}\; + \; \rho^{female}_{A1--A3}) \\ \mathcal{H}_3&: \text{not} \; \mathcal{H}_1 \; \text{or} \; \mathcal{H}_2 \end{align} \]

where \(\mathcal{H}_3\) is the complement and can be understood as neither the first or second hypothesis.

The hypothesis is easily translated to `R`

code

```
hyp <- c("g1_A1--A2 + g1_A1--A3 > g2_A1--A2 + g2_A1--A3;
g1_A1--A2 + g1_A1--A3 = g2_A1--A2 + g2_A1--A3")
```

Note the `g1`

indicates the group and `;`

separates the hypotheses. I again assume the data is Gaussian (although this can be changed to `type = "ordinal"`

or `type = "mixed"`

; see here)

```
test <- ggm_compare_confirm(Y_males, Y_females,
hypothesis = hyp)
# print
test
#> BGGM: Bayesian Gaussian Graphical Models
#> Type: continuous
#> ---
#> Posterior Samples: 25000
#> Group 1: 896
#> Group 2: 1813
#> Variables (p): 5
#> Relations: 10
#> Delta: 15
#> ---
#> Call:
#> ggm_compare_confirm(Y_males, Y_females, hypothesis = hyp)
#> ---
#> Hypotheses:
#>
#> H1: g1_A1--A2+g1_A1--A3>g2_A1--A2+g2_A1--A3
#> H2: g1_A1--A2+g1_A1--A3=g2_A1--A2+g2_A1--A3
#> H3: complement
#> ---
#> Posterior prob:
#>
#> p(H1|data) = 0.13
#> p(H2|data) = 0.825
#> p(H3|data) = 0.046
#> ---
#> Bayes factor matrix:
#> H1 H2 H3
#> H1 1.000 0.158 2.853
#> H2 6.349 1.000 18.113
#> H3 0.351 0.055 1.000
#> ---
#> note: equal hypothesis prior probabilities
```

Note the posterior hypothesis probability for the equality model is 0.825. The Bayes factor matrix then divides those values, for example, \(BF_{21}\) indicates the data were about 6 times more likely under \(\mathcal{H}_2\) than \(\mathcal{H}_1\).

The hypothesis can be plotted

`plot(test)`

It is also important to check the robustness. Here the width of the prior distribution is decreased

```
test <- ggm_compare_confirm(Y_males, Y_females,
hypothesis = hyp,
prior_sd = 0.15)
# print
test$out_hyp_prob
#> 0.18523406 0.74906147 0.06570447
```

which results in a probability of 0.75 for \(\mathcal{H}_2\) (\(BF_{21} = 4.04\)).

Three approaches for testing the same hypothesis were demonstrated in this vignette. This highlights that any hypothesis can be tested in **BGGM** and in several ways.

Rodriguez, Josue E, Donald R Williams, Philippe Rast, and Joris Mulder. 2020. “On Formalizing Theoretical Expectations: Bayesian Testing of Central Structures in Psychological Networks.” *PsyArXiv*.