ols - large effect size and low Rsquare I am a bit puzzled by the relation between the effect size and the $R^{2}$ in a normal ols regression. 
I am fitting a very simple model, with a categorical variable sex (0=man, 1=woman) 
$y = \alpha + \beta1_{Sex}$
The $R^{2}$ of the model is very low (less than 1%) even though the effect size of $\beta1$ (between men and women) is very large and significant. 
Interestingly, when I include more variables in the model, the $R^{2}$ stay very low even though most variables are significant and their effects are large. 
How can I have a model with many significant variables with sizeable effects and have a very very low overall $R^{2}$ ? 
 A: Lets say we are trying to figure out how sex{male=1, female=0} ($x_1$) influences monthly wages ($\hat{y}$). The resulting $R^2=0.05$. The coefficient($\beta_1$) is highly significant ($p<0.001$) so that:
$\hat{y}=\beta_0+350x_1$
This means that amount the surveyed people, there is an average difference in wages between men and women of 350 dollars (men earn more on average). We can say that there is a difference in the population in wages, between men and woman with a $0.1\%$ chance of error (the actual difference can be estimated with confidence intervals, with a predefined margin of error).
The $R^2$ however, tells us that only $5\%$ the variance in wages between individuals can be explained by the difference in their sex. This makes sense of course, as many-many unobserved or confounding elements can influence wages - education, experience, intelligence, field, location etc...
So even if a coefficient is large and highly significant, it might very well still explain but a fraction of the dependent variable.
A: Your model $y\propto Sex$ is simple but the variable $Sex$ does not explain most of the variance in $y$.  In fact, a large p value for the $\beta_1$ coefficient simply indicates that there is a coefficient $\beta_1$ that given the appropriate value will change in the same direction as $y$.  This change explains very little of the variance in $y$ as a function of $Sex$.  If you add just another continuous variable the effect size should be even greater as the fitting now occurs over the range of values of the other variable.  Keep in mind that when you regress on only one categorical variable with values of 0 and 1, when the categorical variable is zero you are fitting against the average of the dependent variable as the product of $\beta_1 * 0 = 0$.  So if there are no more variables you are left with just the intercept and a large error term. As explained in Why is my R-squared so low when my t-statistics are so large? having more observations makes the small explanation of the independent variable by your coefficient more robust but not more precise as your variance remains largely unexplained by the dependent variables.     
