An exponential family has a density of the form
$$
f(y\vert \mu, \phi) = h(y,\phi)\exp
\left(\frac{b(\mu)T(y) - A(\mu)}{d(\phi)}\right)
$$
Above, $\phi$ is the dispersion parameter. Typically you look at models of the form
$$
f(y_i\vert \mu_i, \phi) = h(y_i,\phi)\exp
\left(\frac{b(\mu_i)T(y_i) - A(\mu_i)}{d(\phi)}\right)
$$
where $\mu_i = \vec{x}^\top_i\vec{\beta}$ where $\vec{x}_i$ are your covariates and $\vec{\beta}$ are slopes. Thus, all observations share the dispersion parameter. Now to
Basically I am a bit confused by the meaning of these three concepts: the variance of y (the response), the dispersion parameter and the deviance.
...
Would someone please specify the meaning of the three terms? :)
then the dispersion parameter, $\phi$, is used as the variance is
$$\text{Var}(y_i\vert\mu_i,\phi)=A''(\mu_i)d(\phi) = V(\mu_i)d(\phi),
\qquad V(\mu)=A''(\mu_i)$$
so the dispersion parameter is a parameter used to scale the variance function $V$. Notice that it is used to scale the observations', $i = 1,\dots,n$, variances by the same factor $d(\phi)$.
The variance of $y_i$ is $\text{Var}(y_i\vert\mu_i;\phi)$. It is the variance you expect given a $\mu_i$ and dispersion parameter $\phi$.
The deviance, as Carl writes, is a generalization of the idea of using the sum of squares of residuals. The deviance is given by
$$D = 2\phi\sum_{i=1}^n\left(\log f(y_i\vert \tilde\mu_i) - \log f(y_i\vert \mu_i) \right)$$
were $\tilde\mu_i$ are the linear predictors in a saturated model. To see that this is a generalization of the sum of squares of residuals, we focus on the distribution of canonical form where $b(\mu) = \mu$ and $T(y) = y$. In this case
$$D = 2\sum_{i=1}^n\left(y_i(\tilde\mu_i - \mu_i) - A(\tilde\mu_i) + A(\mu_i)\right)$$
A linear model has
$$A(\mu) = \mu^2/2,\quad \tilde\mu_i = y_i$$
so
$$\begin{split}
D &= 2\sum_{i=1}^n\left(y_i(\tilde\mu_i - \mu_i) - A(\tilde\mu_i) + A(\mu_i)\right) \\
&= 2\sum_{i=1}^n\left(y_i(y_i - \mu_i) - \frac{y_i^2}{2} + \frac{\mu_i^2}{2}\right) \\
&= \sum_{i=1}^n\left(y_i - \mu_i\right)^2
\end{split}$$
which is the usual residual sum of squares.
Best Answer
There are two sides to this question: the mathematical aspect, and the causal (inferential) aspect.
In the mathematical sense, a dependent variable y is a function f of an independent variable x: $y = f(x), f \subset X \times Y$. In this sense, the set X is the set of all players, and the set Y is the set of home run rates. The function f connects the chosen player and his (or her) home run rate. However, this relation can be inversed by use of the inverse function $f^{-1}$, in the sense of $y=f(x) \Leftrightarrow x = f^{-1}(y)$
However, I understand your question as pertaining to the inferential structure. In terms of your question, what depends on what? Clearly, the home run rate depends on the player, because it is a function of some characteristics of the player - his speed, striking prowess, etc. In that sense, the data itself is imbued by a causal structure: if the player changed - say, went into heavy training or used enhancing drugs - he or she could improve performance.
I hope this goes somewhere in answering your question.