If your dependent variable is dummy coded (binary) then you must use a logistic regression for you analysis. There are two types; logit and probit. Both types return very similar results and your decision on which to use is based on personal preference and discipline standards. Economics and marketing tend to use probit while sociology tends to use logit.
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Generally, when the dependent variable appears to be the result of more than one independent variables, a multiple regression model may be suitable. It is difficult to justify adding an additional variable, that does not significantly reduce the residual error of the fit. The setting of thresholds to justify addition of variables is in the area of "stepwise regression." The data must be adequate and consistent with the assumption of independent variables. I note from the first related link: Most authors recommend that one should have at least 10 to 20 times as many observations (cases, respondents) as one has variables, otherwise the estimates of the regression line are probably very unstable and unlikely to replicate if one were to do the study over. See related links. Many more are available in the Internet. Also, many books have been written on the multiple regression- proper and improper use.
You use it when the relationship between the two variables of interest is linear. That is, if a constant change in one variable is expected to be accompanied by a constant [possibly different from the first variable] change in the other variable. Note that I used the phrase "accompanied by" rather than "caused by" or "results in". There is no need for a causal relationship between the variables. A simple linear regression may also be used after the original data have been transformed in such a way that the relationship between the transformed variables is linear.
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Alpha is not generally used in regression analysis. Alpha in statistics is the significance level. If you use a TI 83/84 calculator, an "a" will be used for constants, but do not confuse a for alpha. Some may, in derivation formulas for regression, use alpha as a variable so that is the only item I can think of where alpha could be used in regression analysis. Added: Though not generally relevant when using regression for prediction, the significance level is important when using regression for hypothesis testing. Also, alpha is frequently and incorrectly confused with the constant "a" in the regression equation Y = a + bX where a is the intercept of the regression line and the Y axis. By convention, Greek letters in statistics are sometimes used when referring to a population rather than a sample. But unless you are explicitly referring to a population prediction, and your field of study follows this convention, "alpha" is not the correct term here.