exists with tiie residual variances. For a fraction f of missing data, tiie frac- 

 tional increase in regression coefficient variance attributable to nonrandom 

 missing data is 



Q = 2f/a-n 



The corresponding fractional increase in the residual variance is 



iQ+l){l-f)-l = f 



Locations PAPA, ECHO, Cape St. James, and Langara Island have nonrandom 

 missing data. A correction for such data is reflected in the columns with the 

 asterisks of table 3. The corrections yield more conservative estimates of 

 oq and F. 



The analysis of variance leading to the F-ratio is as follows. Assume 

 that there are N complete years of data. Since the sine and cosine functions 

 yield an integral number of periods per year, the N years of data can be interpreted 

 as 1 year of data with N observations per day. One aspect of this fact is that the 

 5 years of data missing for Cape St. James, as noted in table 1, are not pertinent 

 to the regression analysis. For simplicity, all years are assumed to have 365 

 days. The total sum of squares about the sample mean can be partitioned as 



N 365 365 N 365 



1=1 ]=l j=l i=l j=l 



Where T jj is the observed temperature on the ;-th day of the ?'-th year, and T ■' is 

 the temperature predicted by equation (2) for the ;-th day. For /? = 1, the quantity 



1 r/-n^ 



72 

 / . \ J I 



F=- 



y y(T,-^.-T/)V(365N-3) 



has the F-distribution with 2 and 365N-3 degrees of freedom. Assuming the 

 F-test is robust with respect to missing data, the test is applied in the present 

 situation with 365N replaced by the actual number of observations. 



The least squares method is valid if (1) the error between the true 

 regression curve and the observed value is distributed independently of the 

 independent variables with zero mean and constant variance; and (2) ideally, 

 successive errors are distributed independently of one another. Actually, the 

 problem of using the method of least squares when the error terms are auto- 

 con-elated has been solved if the e's follow certain autoregressive processes.' 



The residuals, after the annual terms are removed, constitute a time- 

 series with another strongly oscillatory terra. If the F-ratios in table 3 were 



20 



