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I am working on a project where I am interested in the following variables:

Dependent variables: $y_1,\, y_2,\, y_3,\, y_4,\, y_5$ (continuous)
Independent variables: $x_1,\, x_2, \ldots,\, x_{18}$ (some continuous, some binary)

The first thing I did was to perform a series of simple linear regressions between each dependent and independent variable.

I found that $x_1$ (continuous variable) was strongly correlated with $y_1, \, y_2,\, y_3,$ and $y_4$, and that no dependent variables were correlated with any other independent variables.

Based on this, I would like to ask the following question:
"Are $\{y_1,\ldots, y_5\}$ related to $x_1$, when including $\{x_2,\, x_3,\ldots , x_{18}\}$ as confounders?"

I can calculate $R^2$ and p-values by performing a partial correlation on each dependent variable, but then:
1) Do I need to adjust the p-values based on multiple comparisons?
2) Does it matter that the dependent variables may be correlated?
3) Is there a more appropriate statistical test to answer my question (like multivariate linear regression)?

I really appreciate the help, I don't have any statistics background.

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1) Do I need to adjust the p-values based on multiple comparisons?

No, this kind of analysis belongs to exploratory analysis. p-value already lost its probability property, and it is just one kind of indicator.

2) Does it matter that the dependent variables may be correlated?

Accosting to your question, you are not interested in the relationship among Ys, so their correlation has no effect on your results.

3) Is there a more appropriate statistical test to answer my question (like multivariate linear regression)?

For each $Y$, fit a linear model with all of the covariates and their interactions that you think they are important, to see what happens (if $Y$ still related to $x_1$.)

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