Trend Estimation: Analyzing Data to Determine Positive/Negative Trend

[Diffy]

Hi, I have a set of data.

What I want to do is say with a 95% certainty this is a positive trend, or it is a negative trend.

Basically I have a set of observations (X_i, Y_i) where i is in {1,2,3,4,5,6...} and X_i = i and Y_i is an observation.

Right now I have the Variance, the standard deviations of my Y_i's and also I have a slope calculated (By Least Squares) from the data set. What do I need to do now to say whether I do or do not have a trend?

Thanks

 

[zli034]

You are using linear regression, therefore the slope is the trend. To test the trend you can test the H0: the slope is 0. Be more specific, this test is to test if the trend is monotonic.

But trend can be some how change direction especially in the time series data. Therefore you need to fit different models to test the specific kind of trend.

 

[Diffy]

Thank you for your reply. How exactly do I test this? Can you find an online example or more information? Or perhaps provide an example or more explanation?

 

[zli034]

Test of of trend is routine practice. For linear trend you need to test the slope.
http://www.ats.ucla.edu/stat/sas/webbooks/reg/chapter1/sasreg1.htm

Here is how SAS do linear regression. The first example has 3 covariates to predict api00, but you only have one covariate, so you need to ignore the other 2. Ignore the 2 only left you acs_k3, which has p-value 0.0553 and it is not significant, therefore the slope of acs_k3 is not significant.

 

[Diffy]

I'm still having trouble. And I am sorry but that article didn't help. I don't have SAS.

From what I understand, I could do a permutation test on my data set. Calculate the slope or trend for each permutation. Then create a distribution. And find where in the distribution my particular trend lies.

Is there a way to do this without actually generating the permutations? It seems like I should be able to calculate this directly somehow.

 

[zli034]

you can search Excel regression to use. Excel regression does t test on slopes. Here is an example Excel did for regression on CPI by year.

year cpi
2001 177.1
2002 179.9
2003 184
2004 188.9
2005 195.3
2006 201.6
2007 207.342
2008 215.303
2009 214.537
2010 218.056

SUMMARY OUTPUT

Regression Statistics
Multiple R 0.990100513
R Square 0.980299025
Adjusted R Square 0.977836403
Standard Error 2.28783132
Observations 10

ANOVA
df SS MS F Significance F
Regression 1 2083.573556 2083.573556 398.0712704 4.15203E-08
Residual 8 41.87337719 5.234172148
Total 9 2125.446934

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -9880.393909 505.1497455 -19.55933661 4.85314E-08 -11045.27131 -8715.516508 -11045.27131 -8715.516508
year 5.025478788 0.251881938 19.95172349 4.15203E-08 4.444637997 5.606319579 4.444637997 5.606319579

 

[Diffy]

zil,

Thanks you for your patience and help. It lead me to the answer I was looking for.

I didn't know it when I originally asked the question but, what I needed to do is perform a t test on the slopes by hand. Using your posts and google's help I have figured out how to do this by hand. Which is exactly what I needed so that I could write this program.

Again thank you very much for helping me find my solution.

-Diffy