Change line type of border

I think this will be very difficult to do with theme options. Even when the panel spacing is 0, I'm pretty sure it's using element_rect to draw the panel borders. Getting the plot you want with theme modifications would require modifying 1 or 2 lines of each panel border with awareness of whether the facet is a left, right, or central facet.

To get around this we can use hline and vline instead. Hacky, but it works:

hline_df = expand.grid(visit = unique(df$visit), yint = c(-Inf, Inf))
vline_df = expand.grid(visit = unique(df$visit), xint = c(-Inf, Inf)) %>%
  mutate(no_dash = !(
    (xint == Inf & visit == tail(levels(visit), 1))  |
    (xint == -Inf & visit == head(levels(visit), 1))
  ))


ggplot(df, aes(x=avisit, y=mean, group=Type, color=Type, shape=Type)) + 
       scale_y_continuous(breaks=seq(0,18,2), limits=c(0, 18)) +
       geom_point(position=pd, cex=2) +
       xlab("") +
       ylab("Mean")  +
       scale_colour_manual(values=c("blue", "red")) +
       scale_shape_manual(values=c("triangle", "circle")) + 
       coord_cartesian(ylim = c(0, 18)) +
       facet_grid(.~factor(visit), scales = "free_x", space ="free_x",switch = "both") +
       geom_hline(data = hline_df, aes(yintercept = yint)) +
       geom_vline(data = vline_df, aes(xintercept = xint, linetype = no_dash), show.legend = FALSE) +
       theme_bw()  +
       theme(axis.text.y = element_text(margin = margin(r = 0)),  
             panel.spacing = unit(0, "mm"),                        
             strip.background = element_blank(),
             legend.title=element_blank(),
             strip.placement = "outside",
             legend.background = element_rect(color="black", fill="white", size=0.5, linetype="solid"), 
             legend.direction = "horizontal",
             panel.grid.minor = element_line(colour="white", linetype="dashed"),
             panel.grid.major = element_line(colour = "white",linetype="dashed"),
             panel.border = element_blank())