Statistical meaning of the ôfö parameter output by Linest 1

[electricpete]

I used linest function to do Multiple Linear Regression of a single independent variable (Y=vibration) against two independent variables (x1 = date, x2 = temperature)

Y = m1*x1 + m2*x2 + b

The output paramters have the following descriptions:
(m2 = slope2) (m1 = slope1)
sd(m2) sd(m1) sd(b)
R^2 sd
f d.o.f.
SS regression SS residual

Here are the numerical values:
0.0013 -0.0027 -52.88529808
0.0004 0.0006 16.28032118
0.88050 0.014
51.6 14
0.0214 0.0029

My question: What is the meaning of f? I vaguely recall that it is something like a goodness of fit test. I think I can find a chart of f in the back of a book, but I don’t remember the meaning.

To my way of thinking, most of what I need to know about the match of the data to the model is told in the other parameters. Since the estimated parameters m1 and m2 are more than 3 times their standard deviation, the variables x1 and x2 probably do contribute to the changes in Y (there is very small probability that the actual slope is 0). R^2=0.88 means 88% of the variability (sum of squares of residuals) is accounted for by the model.

What else can I tell about the results based on f?
Thanks.


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[MintJulep]

Excel's built-in help for LINEST has an explaination of sorts.

 

[electricpete]

I didn't see any explanation regarding "f" in my excel 2003 linest help

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[GregLocock]

The F statistic, or the F-observed value. Use the F statistic to determine whether the observed relationship between the dependent and independent variables occurs by chance.


Assume for the moment that in fact there is no relationship among the variables, but that you have drawn a rare sample of 11 office buildings that causes the statistical analysis to demonstrate a strong relationship. The term "Alpha" is used for the probability of erroneously concluding that there is a relationship.

F and df in LINEST output can be used to assess the likelihood of a higher F value occurring by chance. F can be compared with critical values in published F-distribution tables or Excel’s FDIST can be used to calculate the probability of a larger F value occurring by chance. The appropriate F distribution has v1 and v2 degrees of freedom. If n is the number of data points and const = TRUE or omitted, then v1 = n – df – 1 and v2 = df. (If const = FALSE, then v1 = n – df and v2 = df.) Excel’s FDIST(F,v1,v2) will return the probability of a higher F value occurring by chance. In Example 4, df = 6 (cell B18) and F = 459.753674 (cell A18).

Assuming an Alpha value of 0.05, v1 = 11 – 6 – 1 = 4 and v2 = 6, the critical level of F is 4.53. Since F = 459.753674 is much higher than 4.53, it is extremely unlikely that an F value this high occurred by chance. (With Alpha = 0.05, the hypothesis that there is no relationship between known_y’s and known_x’s is to be rejected when F exceeds the critical level, 4.53.) Using Excel’s FDIST you can obtain the probability that an F value this high occurred by chance. FDIST(459.753674, 4, 6) = 1.37E-7, an extremely small probability. You can conclude, either by finding the critical level of F in a table or by using Excel’s FDIST, that the regression equation is useful in predicting the assessed value of office buildings in this area. Remember that it is critical to use correct values of v1 and v2 computed in the previous paragraph.

Example 5 Calculating the t-Statistics

Another hypothesis test will determine whether each slope coefficient is useful in estimating the assessed value of an office building in example 3. For example, to test the age coefficient for statistical significance, divide -234.24 (age slope coefficient) by 13.268 (the estimated standard error of age coefficients in cell A15). The following is the t-observed value:

t = m4 ÷ se4 = -234.24 ÷ 13.268 = -17.7

If the absolute value of t is sufficiently high, it can be concluded that the slope coefficient is useful in estimating the assessed value of an office building in Example 3. The table below shows the absolute values of the 4 t-observed values.

If you consult a table in a statistics manual, you will find that t-critical, two tailed, with 6 degrees of freedom and Alpha = 0.05 is 2.447. This critical value can also be found using Excel’s TINV function. TINV(0.05,6) = 2.447. Because the absolute value of t, 17.7, is greater than 2.447, age is an important variable when estimating the assessed value of an office building. Each of the other independent variables can be tested for statistical significance in a similar manner. The following are the t-observed values for each of the independent variables.




Cheers

Greg Locock

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