From your description I suspect you are using stepwise regression with ordinary least squares in SPSS. You don't want to do that. See the attached paper for why and some other options you can use. Now to your questions:

1. I would suggest lasso. I assume you are building a predictive model. See the attached papers. I also assume your DV is such that OLS regression is reasonable. See the references in the first paper.

2. assuming OLS the assumptions are given in the attached course notes. If your DV is such that you need other than OLS see the course notes below for common suggestions. Checking the assumptions involves a residual analysis. see:

https://www.google.com/search?q=checking+ols+assumptions&ie=utf-8&oe=utf-8&client=firefox-b-1

3. Prof Roy gave a conservative suggestion. I would say no more than 4.

4. Not much until you correct 1-3 above.

Best wishes, David Booth

Johnathan -

It depends on how complex the relationships are, and how much "noise" (what sigma) you have.  For linear regression with two or three regressors and a model that has some predictive use, you may be fine.  More complex regression may be fine too, depending, as just indicated, upon the fit. 

I recommend graphical residual analysis.  However, overfitting may be a problem, so it is best to hold out some data to use for testing purposes.  With so little data, that may be problematic, but perhaps you could do with n=28 or 29, and use the other 4 or 5 data points (or more?) for testing, to see how well you 'predict' for them.  You might want to research the terms "model selection," and "model assessment," though it is usually expected that you would have a lot more data.  But don't despair yet, you can check it out and see.  Perhaps something in the following will be of use to you:

 https://www.researchgate.net/publication/313925533_Comparing_Model_Performances_Graphically

Also, here is another resource:

 https://onlinecourses.science.psu.edu/stat501/node/279/

Secondly, you want to look for heteroscedasticity, as noted in the second from above linked document.  It is often the case, especially in population statistics. 

Note - You might look through the following, which concentrate on finite populations rather than time series, by the way:  

 https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity

And

 https://www.researchgate.net/publication/324706010_Nonessential_Heteroscedasticity

The maximum number of variables depends upon the regression model fit, as noted above.

Finally, don't worry too much about R^2.   The graphical residual analyses and testing are more important for prediction.

You may also think about "prediction" as compared to "explanation."  The following by Galit Shmueli may be helpful, especially the example in the appendix:

https://www.researchgate.net/publication/48178170_To_Explain_or_to_Predict

I hope that some of the above is helpful to you. 

Cheers - Jim

Hi, recommend you to read - Samuel B. Green (1991): How Many Subjects Does It Take To Do A Regression Analysis, Multivariate Behavioral Research, 26:3, 499-510

PS - When choosing the candidate models whose performance you want to compare, use your subject matter knowledge to make these models reasonable. Note that there are various ways in which your 'independent' variables may actually interact. One is collinearity, meaning you do not want to choose two independent variables which each give you about the same information.

Good comments from our colleague. regards to all of you

Hello

1.How many independent random variables and dependent random variables in your experiment?

2. The sample size and other theoretical requirements for regression analysis and related tests/measures are very important, and cannot be neglected.

2. The regression has five key assumptions:

. Linear relationship;

. Multivariate normality;

. No multicollinearity;

. No auto-correlation and

. Homoscedasticity.

3. In order suggest a method for analysis, number 1 above needs to be explained.

4. For more details, please refer to:

. Richard Johnson and Dearn Wichern (2007). Applied Multivariate

Statistical Analysis. Chapter 4, pages: 149-208.

.Joseph Hair, Wuilliam Black, Barry Babin, Rolph Anderson and Ronald . Tatham (2006).Multivariate Data Analysis. Chapter 4, pages: 237-243.

. Norman R. Draper, Harry Smith (1998). Applied Regression Analysis, 3rd Edition, Wiley.

. Snedecor, W.G. and Cochran,W.G. (1989). Statistical Methods, Blackwell.

Regards,

Zuhair

In addition to my last answer, you can use SPSS to analysis your data. Please refer to:

. Landau, S. and Everitt, B. S.(2004). A Handbook of Statistical analyses using SPSS.

.Howitt, D. and Cramer, D.(2008). Introduction to SPSS.

Regards,

Zuhair