Linear Regression Analysis Example from Zar (1999)

Biostatistics BI 45, Saint Anselm College

 

Step 1) State the biological question.

Is there a relationship between age(days) and wing length of sparrows?

Step 2) Translate the biological question into statistical hypotheses.

Step 3) State the alpha level.

Step 4) State the statistical test and the number of tails.

Because we are trying to determine whether there is a relationship between age and wing length we will use a linear regression analysis. This is a two-tailed test.

Step 5) State the assumptions of the test.

The subjects in each sample must be randomly selected from a population. There must be normal distribution of Y for each value of X. There must be no error in each value of X.

6) Build the linear model and test it

AGE WING D D CROSS PROD x^2 of D x^2 of D
  LENGTH FOR AGE FOR WINGL   FOR AGE FOR WINGL
3 1.4 -7 -2.015 14.105 49 4.06
4 1.5 -6 -1.915 11.49 36 3.667
5 2.2 -5 -1.215 6.075 25 1.476
6 2.4 -4 -1.015 4.06 16 1.03
8 3.1 -2 -.315 .63 4 .099
9 3.2 -1 -.215 .215 1 .046
10 3.2 0 -.215 0 0 .046
11 3.9 1 .485 .485 1 .235
12 4.1 2 .685 1.37 4 .469
14 4.7 4 1.285 5.14 16 1.651
15 4.5 5 1.085 5.425 25 1.177
16 5.2 6 1.785 10.71 36 3.186
17 5 7 1.585 11.095 49 2.512
             
Sum 44.4 130   70.8 262 19.657
n 13 13        
Mean 3.415 10        
df 12 12        

 

6a) Linear model

 

 

6b) Test the model - F test

 

 

Find the actual P value

F.001(1)1,11=19.7< F.0005(1)1,11=23.7<Fobs=388.68

P<.0005

 

6c) Test the model - t test

 

 

Find the actual P value

t.002(2)11=4.025< t.001(2)11=4.437<tobs=20.00

P<.001

 

 

Step 7) Reject the null hypothesis

 

Step 8) Interpret the results of the analysis as it relates to the biological question raised in 1).

 

We rejected the null hypothesis in Step7. There is a relationship between wing length and age (t=20.00,df=11, P<0.001) or (F=388.68, df = 1, 11, p<0.0005).

 

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Trademark and Disclaimers

Copyright © 2001 Jay Pitocchelli. All rights reserved. The contents of this page are the intellectual property of Dr. Jay Pitocchelli for distribution to students enrolled in Biostatistics BI 45 at Saint Anselm College. These pages may not be copied, photocopied, reproduced, translated, or published in any electronic or machine-readable form in whole or in part without prior written approval of Jay Pitocchelli. Students enrolled in Biostatistics BI 45 at Saint Anselm College have permission to print this material for their lecture notes. All formulae and critical values from: Zar, J. H. 1999. Biostatistical Analysis. (4th ed.). New Jersey, Prentice Hall.