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With the Data Analysis command in Excel I made a plot, which I can't post, because this is my first post ever.

X - abortions, Y - maternal deaths. 
SUMMARY OUTPUT is this:                 

Regression Statistics                               
Multiple R  0.796827692                         
R Square    0.634934371                         
Adjusted R Square   0.612117769                         
Standard Error  9.360700845                         
Observations    18                          

    df  SS  MS  F   Significance F          
Regression  1   2438.341925 2438.341925 27.82773596 7.54568E-05         
Residual    16  1401.963525 87.62272032                 
Total   17  3840.30545                      

    Coefficients    Standard Error  t Stat  P-value Lower 95%   Upper 95%   Lower 95.0% Upper 95.0%
Intercept   16.80848279 8.155297195 2.06105092  0.055938494 -0.479974838    34.09694041 -0.479974838    34.09694041
X Variable 1    0.000645551 0.000122375 5.275200087 7.54568E-05 0.000386128 0.000904973 0.000386128 0.000904973

RESIDUAL OUTPUT                             

Observation Predicted Y Residuals                       
1   65.10793207 -0.157932075                        
2   82.68111012 -7.861110124                        
3   76.14620154 -14.17620154                        
4   71.93721172 -3.117211723                        
5   68.91926273 7.810737266                     
6   66.32285828 3.197141723                     
7   60.35861641 16.68138359                     
8   60.13396481 19.00603519                     
9   58.92678521 7.643214785                     
10  56.63572619 5.004273815                     
11  56.81002484 -7.890024844                        
12  52.19433815 -0.254338154                        
13  49.92070899 -7.980708991                        
14  47.42630153 -10.79630153                        
15  45.88472673 -5.024726728                        
16  44.74791215 0.982087854                     
17  43.49489846 4.025101541                     
18  40.40142005 -7.091420051    

Question is: how to test all assumptions (e.g. outliers, normality, homoscedasticity) of the regression analysis? Can I figure out those with Excel? I also just downloaded Mathematica software - so, I am not familiar with it, but would like to use it if necessary. If you could help me with detailed steps?

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Hi Aliya, welcome to our site! Although we welcome all statistics-related questions, we do not operate as a generator of textbooks nor as personal tutors, which is what your question appears to demand: to formulate an acceptable question you need to focus it on a specific, practical problem; state what progress you have made; and identify precisely where you need help. For more information about how to write a good question, please consult our faq. –  whuber Aug 28 '12 at 14:32
@Whuber I guess I disagree.The practical problem is "How do I test for normality"which seems to me to be precise enough.My answer reflects that in my opinion he used a solution technique (OLS) that was like a pop-gun rather than the howitzer he needed to ferret out any dynamic structure or unspecified structure due to unknown assignable causes at periods 12,4,9 and 8. Identifying the "true cause" of these anomalies leads to enhanced understanding and at the same time rendering the error process to white noise.But that's just my opinion.Does anybody else on this list agree with meor am I alone? –  IrishStat Aug 28 '12 at 14:54
You're addressing questions that aren't even explicitly set out here, @Irish. This one asks about testing "all assumptions" as well as "detailed steps" with Excel and Mathematica. There is no evidence that this is a time series problem. –  whuber Aug 28 '12 at 15:39
@whuber I answered the question because even though it asked about all the diagnostics related to regression I thought that I could give concise answers on what methods are used. As far as providing detailed steps for carrying this out in Excel or Mathematica that probably is not appropriate for a question. IrishStat provided a detailed description of how to deal with time series modeling assumptions. I think it is interesting even though I agree that the OP was probably not looking at time series. –  Michael Chernick Aug 28 '12 at 15:51
@huber the acf of .56 for lag1 in his residuals suggests time series data or evenly spaced spatial data .... at least that's what I think. I can't contrive an alternative state of nature that would create a set of residuals with this acf. –  IrishStat Aug 28 '12 at 16:05
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closed as not a real question by whuber Aug 28 '12 at 14:32

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2 Answers

To follow up on Michael's remarks. You didn't say whether or not the original data was time series data but an analysis of the residuals can often uncover that. Is this data time series i.e. observations taken over time for a particular country/section or is data taken for 1 period of time for many countries/sections?. My analysis tells me that this is most probably time series data.

Concluding about normality requires testing for independence. On the off chance that you had misused regression and employed an OLS model, I took your 18 residuals into AUTOBOX to investigate the need for ARIMA structure and the detection of unspecified deterministic structure like Pulses, Level Shifts, Seasonal Pulses and/or local time trends. In the absence of knowledge about the possible frequency of measurements, I assumed that the data was annual. Here I plot the ACF of the original series ( your residuals ). enter image description here suggesting a time dependence structure in the original series. A plot of AUTOBOX'S model residuals suggests randomness. enter image description here . The model for your 18 values is enter image description here which includes an AR(1) coefficient and a few pulses. Also, the statistical summary of this model is of interest as it suggests the importance of the model identified enter image description here . The actual/fit/forecast graph illustrates the analysis visually. enter image description here while the cleansed graph pinpoints the unusual activity enter image description here. In summary, data exploration/assumption checking/data mining/exploratory data analysis all comes seamlessly together in this simple analysis. My conclusions are that your residuals are not normally distributed ( see histogram )enter image description here and your conclusions about statistical significance of your regression model coefficients are suspect. Auto-correlation in your error series is a de facto example of heterogenous variance and in this example is simply handled by the extraction.identification of the embedded ARIMA process and a few anomalies. There is no need for over-complicating the solution by non-linear transformations. The mantra of the statistician should be KISS (but not too simple ! ) and do no harm.

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I do not know anything about the particular software you have, but I can talk about methodology that can be used. In regression you can look for outliers by testing for points that exert high influence on the regression parameters. This is done by computing the influence function which essentially looks at how much the estimate changes when the data point in question is removed. Normality can be checked graphically by generating qq plots of residuals and looking for departures from a straight line. Goodness of fit tests on the residuals such as the Shapiro-Wilk test can formally test for normality.

One way to look for change in variance would be to estimate the varaince of the residuals in one portion of the x space and test equality of variances by comparing it with the estimate is a different region of the x space. An F test can be applied.

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