<THE FOOLISH FOUR>

A one-tailed T test
by Ann Coleman
([email protected])

Reston, VA (November 11, 1998) -- I woke up this morning chanting, "A one-tailed T test. A one-tailed T test." I couldn't get it out of my head. I go so desperate I started singing "It's a small world after all...." It's nice that they let me have my laptop in here, but now the other basket-weavers want one, too.

Yes, friends, it's more statistics today, but this is it for a while. Today we look at a test similar to the one discussed on Friday, but more sensitive for this kind of data, to see how the claim that our strategies beat the market holds up.

First, though, I have to correct an error that I made in Friday's article. Luckily, when corrected, the confidence level for the 37 year test of our strategies increases slightly. Beginner's luck, I guess. What happened was I missed copying the top two rows of data (1961 and 1962) into my table, but mathematically I was still assuming I had 37 years. Really dumb, eh? That will teach me to do statistics on deadline. (Of course, one could have concluded that that was a bad idea before making a fool [small f] of oneself in public.)

Rather than cluttering up this space with more numbers, those of you who are interested in seeing all the details can click over to our web message board and study the results. The general conclusion is that for both the 37-year and 25-year periods studied, the High Yield 10 shows a positive correlation that is "suggestive" but not statistically significant, while the Foolish Four outperforms the Standard & Poor's 500 Index at the 95% confidence level (the probability that the outperformance is due to chance is less than 5%) and the RP outperforms the S&P 500 at the 97% confidence level (the probability that the outperformance is due to chance is less than 3%).

Today's calculations (done yesterday and checked obsessively today) are for a more specific test that many readers suggested would be more appropriate for this type of data. It is called a paired T Test, and it's really kind of cool. Instead of just looking at the averages of the yearly returns like we did Friday, in this test you look at how one strategy compared with another on a year-by-year basis. Then you look at the average of those comparisons. The test tells you whether or not the differences you are seeing each year are due to random differences -- after all, no two strategies will perform the same all the time -- or if there is some underlying factor that is causing those differences.

In this case, we want to know if the strategies tested are outperforming the S&P or if the observed differences are just due to chance. Because we are specifying that the differences are positive (no one is suggesting that they could possibly be underperforming the market, are they?), the test is called a one-tailed test. You are looking to see if your test results fall in the tail end of the bell curve on the right side (positive correlation). A two-tailed test looks for either a large positive difference or large negative difference, i.e., the results can be in either the right tail or the left tail of the bell curve.

OK. I'm putting the textbook away now.

Imagine you are at a fishing camp. Every day, two fishermen go out on their own and come back with a fish. One guy usually gets a bigger fish. You wonder: Is John just lucky, or is he a better fisherman? There are a couple of ways you could check that out. You could just average the weights for all of John's fish and compare that with the average weight of all of Jack's fish, like we did last week. That confirms that John's fish are bigger and the z test that we did would tell you how likely it is that the difference is due to luck.

A better test for this situation, where the data is "paired" -- i.e., they are going out on the same day, in the same weather conditions, in the same general area, etc. -- is the paired T Test. To do that, you would weigh the fish and record the difference each day. Day 1 John's fish is 6 ounces bigger than Jack's. The next day, Jack's fish is 2 ounces bigger than John's, etc. Then you look at the average difference in each day's catch. That's what we did with our T Test. Because we are assuming that our strategies outperform the market, we use the one tailed test.

The full details are posted on our message boards. You can see all the numbers by clicking here. Followers of the High Yield 10 (Dogs of the Dow) strategy will be happy to see that this more sensitive test does pick up on the better performance of the Dogs. For the 37-year sample, the difference between the Dogs and the S&P was significant at the 97% level. The Foolish Four outperforms the S&P 500 at the 99% confidence level, and the RP4 outperforms at the no-way-in-hell-is-this-due-to-chance confidence level of 99.9%. For the 25-year data, the confidence intervals are slightly higher across the board. (No surprise there -- the sixties was not a good decade for any of the Dow strategies.)

I also look at two other pairings: The Dow vs. the S&P 500 and the Foolish Four vs. the RP4. As would be expected if this test were valid, the difference between the Dow and the S&P is not significant -- in fact, it is almost nonexistent. The RP4 strategy shows a positive correlation that is not statistically significant. For the past 25 years, the probability that the RP4's outperformance is due to chance is about 25%.

Now, the important question. What does all this really mean? Well, in a way, it doesn't tell us anything we didn't already know. The strategies we follow here do better than the market. We knew that already, but now we know it in a way that may be somewhat more defensible to critics.

An even more important question may be, what doesn't this mean? It doesn't mean that the strategies will beat the market this year or next. It doesn't mean that they will continue to perform as well in the future as they have it that past (but I am looking into something that might shed some light on that question). It doesn't even mean that you should invest in the Foolish Four. That, as always, is up to you. And it still doesn't mean that the stock selection strategies we employ are the cause of the outperformance, although it is not unreasonable to assume that, as long as you realize it is your opinion, not some kind of statistically proven fact.

However, I feel sure that we can safely conclude that we have a rigorous mathematical proof that somethin's goin' on here!

Fool on and prosper!

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Current Dow Order | 1998 Dow Returns

11/11/98 Close

Stock Change Last -------------------- UK - 3/8 43.13 IP -1 1/2 44.44 MO - 1/4 53.88 EK - 13/16 77.75

Day Month Year FOOL-4 -1.35% 2.85% 14.07% DJIA -0.45% 2.70% 11.58% S&P 500 -0.64% 2.03% 15.51% NASDAQ -0.19% 5.12% 18.58% Rec'd # Security In At Now Change 12/31/97 206 Eastman Ko 60.56 77.75 28.38% 12/31/97 276 Philip Mor 45.25 53.88 19.06% 12/31/97 289 Int'l Pape 43.13 44.44 3.04% 12/31/97 291 Union Carb 42.94 43.13 0.44% Rec'd # Security In At Value Change 12/31/97 206 Eastman Ko 12475.88 16016.50 $3540.63 12/31/97 276 Philip Mor 12489.00 14869.50 $2380.50 12/31/97 289 Int'l Pape 12463.13 12842.44 $379.31 12/31/97 291 Union Carb 12494.81 12549.38 $54.56 CASH $754.73 TOTAL $57032.54