{"id":320,"date":"2018-11-30T12:00:28","date_gmt":"2018-11-30T10:00:28","guid":{"rendered":"http:\/\/benediktehinger.de\/blog\/science\/?p=320"},"modified":"2018-12-01T12:09:20","modified_gmt":"2018-12-01T10:09:20","slug":"interaction-and-effect-sum-coding","status":"publish","type":"post","link":"https:\/\/benediktehinger.de\/blog\/science\/interaction-and-effect-sum-coding\/","title":{"rendered":"Interaction and Effect\/Sum Coding"},"content":{"rendered":"

Some time ago I wrote a blog post on dummy & effect coding<\/a>. I made some new plots to visualize why the interaction in sum\/effect is coded as it is.<\/p>\n

Let’s take a typical 2×2 design. We have two 2-level factors $A$ and $B$ and we also allow for an interaction.
\n$$y \\sim A + B + A:B$$<\/p>\n

We code A with -1 \/ 1 and B with -1 \/ 1 (depending on the level e.g. On=1, Off=-1)<\/p>\n

The interaction is coded as the multiplication of A and B: $A * B$. Therefore if $A$ and $B$ are both in the same level (both “off” or both “on”) we get a $+1$, else a $-1$.<\/p>\n

Side remark: This is different in dummy\/reference coding, where the interaction only codes what is extra if both A & B are “on” (turns out that the magnitude of the interaction is just double – but this is a story for another time).<\/em><\/p>\n

\"\"<\/a>
\nIn the first figure I added the main effects of $A$ and $B$ as Blue and Purple lines. The main effects in reference coding are relative to the means of the group means.<\/p>\n

In order to model the original data points, one needs to add the main effects and the interaction together:
\n\"\"<\/a><\/p>\n

Notice that the way we have to add the interactions and main effects is exactly the multiplication I introduced earlier. That is, if we need to take -1 for $A$ and +1 for $B$, you bet we will need -1 for $A:B$.<\/p>\n

One way that I like to think about the interaction in effect coding is to think “What would be my prediction if there would be no interaction?”.<\/p>\n

\"\"<\/a><\/p>\n

“What if there would be a model without interaction” is marked in black (it’s only using the main effects!). Note that the two black lines are parallel. Adding the red interaction-lines again helps us to move to the original datapoints.<\/p>\n","protected":false},"excerpt":{"rendered":"

Some time ago I wrote a blog post on dummy & effect coding. I made some new plots to visualize why the interaction in sum\/effect is coded as it is. Let’s take a typical 2×2 design. We have two 2-level factors $A$ and $B$ and we also allow for an interaction. $$y \\sim A + B + A:B$$ We code A with -1 \/ 1 and B with -1 \/ 1 (depending on the level e.g. On=1, Off=-1) The interaction is coded as the multiplication of A and B: $A * B$. Therefore if $A$ and $B$ are both in…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-320","post","type-post","status-publish","format-standard","hentry","category-blog"],"_links":{"self":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/posts\/320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/comments?post=320"}],"version-history":[{"count":0,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/posts\/320\/revisions"}],"wp:attachment":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/media?parent=320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/categories?post=320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/tags?post=320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}