Student's T-Test For Paired Data
Hi PJ,
Something strange happened when primary data were converted into percentages. Look at the ratios between paired primary data and those between paired percentages and note that none of the former exceed unity whereas some of the latter do.
Pair Primary %
1 0.970 0.969
2 0.999 1.016
3 0.994 1.005
4 0.992 0.993
5 0.998 1.006
6 0.998 1.003
7 0.981 0.982
Actually, the sign test could be applied by assigning a minus to ratios below unity and a plus to those above unity and looking up the probabilities associated with the plus and minus counts.
I decided to apply the more robust t-test to the set of paired primary data and found out that the calculated t-value of 2.666 for the difference of 83 units between means of 9251 and 9334 units exceeds the tabulated value of t0.05;6=2.447 so that these means differ significantly. The Bias Detection Limit (BDLs) for the Type I risk only is +/-76 units, and becomes +/- 137 units for the combined Type I and Type II risks. BDL(I)s and BDL(I&II)s are effective control and action limits for control charts.
The variance of differences is equivalent to a coefficient of variation of 0.89%rel which implies a high degree of precision for the applied measurement procedure.
Do not apply the t-test to paired percentages. Send me your email address and I'll transmit an Excel file with the t-statistics for paired primary data.
Kind regards,
Jan W Merks ■
Re: Paired T Test
For interest sake I converted your information back to the supposed totals and thenlooked at them:
Set 1 Set 2
19810.53 19795.92
19482.08 19826.96
19470.85 19671.17
19653.21 19664.69
19589.02 19748.57
19673.59 19764.88
19567.36 19585.24
Paired t 0.05489029
Which is greater than the individual. ■
Student's T-Test For Paired Data
Hello Mike,
Your conversion filled my head with ideas but I don't know what they are. PJ's sets of primary data and percentages display coefficients of variation of 25% whereas your "supposed totals" give a coefficient of variation of 0.6%! So what gives?
I also wonder how you looked at the "supposed totals" because the calculated t-value of 0.055 between the means of Set 1 and Set 2 is incorrect. I should point out that tabulated values of the t-distribution are given with 3 decimals. Is it perhaps possible that the "supposed totals" carry too many decimals too?
Kind regards,
Jan W Merks ■
Re: Paired T Test
I took his data (7528) and divided by the percentage/100 (38.0/100) to generate the original total (7528/(38.0/100) =19810.53). If 7528 is not 38.0% of 19810.53 then what is the relationship between 7528 and 38.0%? ■
T-Test For Paired Data
Hi Mike,
Perhaps PJ could clarify whether or not his "raw data" are indeed his original primary data, and how he converted raw data into percentages in such a manner that some ratios were inverted from less than unity between raw data to more than unity between percentages, and the reverse. Surely, such reversals are bound to impact the t-statistics!
Kind regards,
Jan W Merks ■
Paired t test
Hi,
Just a quick question, can the paired t test be used on pecentages, and if so why would this give a different significance than if done on the raw figures.
the data is :
7528 - 7760
10871 - 10885
8684 - 8734
13148 - 13254
10343 - 10368
6630 - 6641
7553 - 7697
shows p=0.037
as %'s
38.0 - 39.2
55.8 - 54.9
44.6 - 44.4
66.9 - 67.4
52.8 - 52.5
33.7 - 33.6
38.6 - 39.3
shows p=0.648 ■