multiplikativ og additativ effekt

[tetvis]

Hva er egentlig forskjellen på om de ulike forklaringsvariablene har
en multiplikativ eller additativ effekt på responsvariablen. Når vi
holder på med de linealiserte modellene så har forklaringsvariablene
additativ effekt? Men når snakker vi da om mulitplikativ effekt?

[Jisca]

With an additive effect, a change in the explanatory variable adds
something to the response variable, while with a multiplicative effect
the response variable gets multiplied by something. For example, with
the helicopters, in the first case wings-up vs. wings-down would
increase flight time by 3 seconds, while in the second case it would
make it 1,5 times as fast. You get the multiplicative effect because y
= exp(a + b*x) can be re-written as y = exp(a) * exp(b*x).

[ego]

Men er det noen måte man kan lese dette ut fra summary, om effekten av
en variabel er additativ/multiplikativ? og kan samme modell ha både
additative og multiplikative ( i R)?

[Jisca]

It is possible to have both additive and multiplicative effects, but
not in the models that we used. If the response variable is log
transformed, the effects on the untransformed variable will be
multiplicative. Without log transformation, the effects are additive

Maybe it's easiest to explain with some examples. Imagine x is a
factor with 2 levels, and x1 and x2 are dummy variables (x1=1 and x2=0
for the 1st level, and x1=0 and x2=1 for the 2nd level), and the
response variable y is log transformed (or you used a log-link
function).

log(y) = a + b1*x1 + b2*x2
y = exp(a + b1*x1 + b2*x2) = exp(a) * exp(b1*x1) + exp(b2*x2)

Thus, in case of the first level, you get
y = exp(a) * exp(b1*1)
and for the second level
y = exp(a) * exp(b2*1)


If you have a continuous variable x, the interpretation of the summary
table is easiest when this x variable is log-transformed as well, thus
log(y) = a + b* log(x)
y = exp[a + b*log(x) ] = exp(a) * x^b

So different values of x results in multiplying y relative to the
'base line' exp(a).

Hope this cleared things up a bit?