# blueollie

## Once again, all over the place: videos, denial, mammograms

Workout notes Treadmill: 6 mile run in 1:02:50. Started off at 11:0x mpm and did 2 minutes each in the following pattern: 0-.5-1-1.5-2 then 10:42 (same pattern) then 10:31 for most of the rest: 0-.5-1-1.5-2-2-1.5-1-.5-0 then 5 minutes each at 2-1.5-1 then I finished the rest at .5, increasing the pace each minute.

Then 2 miles (16 laps of lane 3) of walking in 29:37 (14:23 for the last mile).

What I’ve noticed: while my legs aren’t classically “dead”, it is almost as if someone sucked out my quad muscles with a straw. They are, well, not doing a thing.

Posts
Physical Stuff

Since we are talking gym: this “gym stereotype” clip is funny. I am the old man in the locker room; I suppose that comes from the fact that many of us don’t look at others…so what is the fuss? It just doesn’t register any more.

Now for some physical craziness. Yes, the law-and-order person in me wondered if these people had the proper permissions to do this. But, well, the video is rather incredible. Physically, these guys are much of what I am not.

Science
Evidence based medicine and science is hard. We create models and then go with our best educated guess…and sometimes it takes years to gather data. Here is a vast study about mammograms and their effectiveness:

One of the largest and most meticulous studies of mammography ever done, involving 90,000 women and lasting a quarter-century, has added powerful new doubts about the value of the screening test for women of any age.

It found that the death rates from breast cancer and from all causes were the same in women who got mammograms and those who did not. And the screening had harms: One in five cancers found with mammography and treated was not a threat to the woman’s health and did not need treatment such as chemotherapy, surgery or radiation.

The study, published Tuesday in The British Medical Journal, is one of the few rigorous evaluations of mammograms conducted in the modern era of more effective breast cancer treatments. It randomly assigned Canadian women to have regular mammograms and breast exams by trained nurses or to have breast exams alone.

Researchers sought to determine whether there was any advantage to finding breast cancers when they were too small to feel. The answer is no, the researchers report.

Unfortunately, this study will probably be pillared by those whose lives were saved, so they think, by mammograms. Remember: this is NOT a study about regular breast exams; it is about mammograms which are supposed to catch the cancer at the early stages.

So, someone who had a genuine harmful cancer detected by a mammogram and was saved may have well be saved by a later detection via a conventional exam.

I suggest reading the whole article; much of the data that shows “x out of 1000 were saved by mammograms” came out before the newer drugs came out.

I don’t know what to think because this isn’t my field of expertise. But it is interesting, to say the least. I just hope that science and statistics determines the best policy and not emotion.

Now about statistics and onto politics: remember the morons and their “unskewed Presidential race polls”? Well, these people haven’t learned a thing; they are refusing to believe the current data about the Affordable Care Act.

I suppose that instead of breaking people down by “conservative/liberal”, we should break them down by “convinced by evidence/not convinced by evidence”.

Social Views Did you know that people who won lotteries changed their economic views in the conservative direction? Now there are some caveats in this study (e. g. people who are likely to play a lottery might have a different mentality that those who don’t; and yes, the lottery really is a tax on those who can’t do math). But Paul Krugman has a ton of fun with this finding.

February 14, 2014

## Stock market graph pattern

I saw an article which posted this graph:

Oh, the text admits that the scales aren’t the same (left versus right) though the time lines are. This is supposed to mean something?

Well, I took the liberty to look at longer trends (1922 to 1930 and then 2007 to 2014) (using this tool)

I don’t see a whole lot of similarity.

Looking at 2 years (as in the “scary graph”):

Hmmm, I suppose that this 1935 to 1937 could be made to fit too.

And one can look at the various cycles, this time scaled in percentages:

I suppose it is human to recognize patterns even where none exists.

Now, I am not a market expert; there might be other signs of a impending crash. But I kind of doubt that pattern fitting is a legitimate sign.

February 12, 2014

## Contempt for elementary education and other topics

Workout notes
Shorter weight workout followed by a cold 4 mile road walk (Bradley Park hill course). It was cold (15 F, or -9 C), somewhat breezy and sunny; there were isolated 50 to 100 meter stretches that were completely “frozen snow/ice” covered. But I wanted to get outside a bit.

The weight workout was a bit different today: part of the rotator cuff (dumbbells), hip hikes, Achilles:
pull ups: 15, 15, 10, 10 (good)
super set with dumbbells: 3 sets each of:
seated military (sets of 12 with 50′s)
upright rows (sets of 10 with 25′s)
bench presses (sets of 10 with 70′s)
bent over rows (sets of 10 with 65′s)
curls: (sets of 10 with 30′s)

Then an ab super set; 3 sets of 10 with crunch, twist, sit back, vertical crunch.

Then came the outdoor walk.

Posts of the day
The NSA sometimes put tracking/control devices in computers that were going overseas; hence they could easily spy on or manipulate computer activity.

Fun with statistics:
Of course correlation and causation are not the same. Then again, sometimes there are good reasons for a non-causal correlation (e. g. my time to run the mile slowing down with years of marriage or the years that Obama has been in office) and sometimes the correlation is simply spurious. Here is a “fun” collection of them.

Oh yes, sometimes there is a valid correlation but the cause and effect are reversed: for example basketball players tend to be tall. So, your “how to get taller” program involves getting your client to take up basketball.

Educational matters

Some time ago I remember seeing a poster outside of a student affairs office; I believe the poster had a picture of various women yelling at a man in the middle; one of the things being said by the females was “how we dress has nothing to do with sex.” Really? Check this out. This is about a sorority “twerk off”.

So, now we’ll hear stuff about “sexualization” and…oh yes, “slut shaming”. Seriously.

My view: this twerking contest is young people being, well, young people. It is all part of the human mating ritual. It neither surprises nor outrages me. No, these women aren’t doing this for me or with people like me in mind; for me, “twerking” is, say, my wife bending over to get her pills out of the lower cabinets or bending over in the garden, etc.

Our society is too tense about these matters, IMHO. The only thing that I ask: if this is going to end up in “new kids”, make sure that you can SUPPORT those kids BEFORE having them, ok? I am not a conservative, but the old saying “you breed ‘em, you feed ‘em” makes sense to me.

And speaking of kids, they need to be educated too.
In the local paper, there have been a series of articles about cheating on standardized tests for “special needs” students. Here is one such article:

■ “Charter Oak staff violated ISAT testing protocol in providing inappropriate testing accommodations to special education students during the administration of the ISAT.” Teachers directed students to correct answers in a variety of ways, going as far as to erase answers themselves.

■ “All staff members interviewed reported they did not receive any formal training on ISAT administration on a yearly basis.”

So, they appear to be saying “I’m sorry we cheated, but we weren’t trained enough to know that changing the pupil’s answers or erasing their wrong answers was cheating.” You need to be TRAINED to know that is wrong?

The proud parents who attended Lincoln Elementary’s honor roll assemblies years ago assumed the school was a shining example of academic achievement.
Kids by the dozens lined up to be celebrated for earning grades that put them on the honor roll.

Then the school in St. Charles got state test results.

Most of the students failed, casting doubt on the school’s success and challenging the validity of many of its students’ glowing report cards. Administrators knew they had a problem.

What they did next upended everything parents, teachers and students thought they knew about grading.

St. Charles joined a national movement that — sometimes amid a formidable backlash — is rebuilding how a child’s performance in a class or course is calculated.

It’s a switch that seeks to move away from rewarding students merely for completing work, and instead bases grades on mastery of a subject.

Swept away are points for finished homework assignments, or good behavior and class participation. Instead, grades are more heavily based on exam results and the quality of work.

Oh my goodness: you mean making a good grade in the subject should infer having some demonstrated ACTUAL KNOWLEDGE of the said subject???? Who knew?

But reading this was useful to me. Some time ago, a “business calculus” student came up to me in anguish. She showed me her homework paper with 0 points on it. She said “I did all this work here, and it was marked WRONG.” I said: “yes, it was marked wrong because the “work” was totally incorrect; there was no correct work here. She gave me the “are you serious?” look; it was if having to be correct to get credit was a new concept for her.

Maybe this is why?

So, none of this is flattering to our grade school educators or educational system. But….yes, I know, this isn’t ALL school districts; these aren’t ALL of the educators and yes, much of the blame might be put on what happens to the pupil BEFORE they get to school (at home) and on this as well:

So yes, I know that there are good, dedicated teachers and educators who are busting their rear ends to do something about it, and these people need good pay and our moral support.

January 15, 2014

## Some halftime stats and science

Yes, I am blogging at halftime of the Oklahoma versus Alabama game. OU leads 31-17 but Alabama has the type of team that can overcome adversity…and I remember the Chick-Fil-A Peach Bowl where Duke lead 38-17 at the half only to lose to the (ugh) Aggies.

The quality of this blog has suffered recently due to…well, increasing business. First it was the super busy semester and then it was vacation.

Hopefully, I can talk about a few things of substance this time.

Weather: yes, it is very cold in Illinois this “winter”. The jet stream has dipped and we are paying the price as the Jet Stream holds back the Arctic Air Mass.

Now of course, Republicans deny global warming…and now an increasing number are denying evolution:

There also are sizable differences by party affiliation in beliefs about evolution, and the gap between Republicans and Democrats has grown. In 2009, 54% of Republicans and 64% of Democrats said humans have evolved over time, a difference of 10 percentage points. Today, 43% of Republicans and 67% of Democrats say humans have evolved, a 24-point gap.

Paul Krugman says that this reflects increasing tribalism (“what does a good conservative believe?”) which, of course, has consequences in other public policy matters (e. g. macroeconomics). Hence Republican candidates have to be very careful not to present the unvarnished truth if they want to keep their base (e. g., Mitt Romney walking back his statements about cutting spending during a recession limiting growth)

Now, there is peril for liberals here too: this is one reason those of us who are scientifically literate must speak out for science, even when it goes against what many of our liberal political allies might think:

What this tells us is that elite opinions matter a lot in public discourse. The gap between liberals and non-liberals is not really there on this issue (GMO) at the grassroots. That could change, as people of various ideologies tend to follow elite cues. This is why the strong counter-attack from within the Left elite is probably going to be effective, as it signals that being against GMO is not the “liberal position.”

The same applies to woo-woo, “alternative medicine”, the irrational attacks against “fracking” (some attacks about it being improperly or inappropriately used ARE legitimate), etc.

I don’t want liberal leaning media to be at the point where it makes the reader more ignorant than before; here is an example of the Wall Street Journal doing exactly that (on income inequality).

Aging and time to failure curves
It is well known that as we age, the probability of dying in a given year goes up. In fact, the probability of dying in a given year doubles with every 8 years of life. Example: if you are married to someone who is 16 years older than you are, they are 4 times more likely to die in a given year than you are.

This article discusses the various mechanisms of why this might be true; it makes for interesting reading.

The bottom line: the model of the attacks on the body being produced at a constant rate, but the body’s ability to fight those attacks being reduced at a linear rate DOES fit this model.

Now as far as the bathtub curve, the lead in to this reliability engineering blog post gives a nice introduction to it, though this article deals with how current reliability engineering deals with “burn in failures” and how “time to obsolescence” affects the curve.

January 3, 2014

## Embarrassment on the Hike and Bike trail

The good news: I enjoyed my run. I did have to stop once to smooth the tongue in my shoe (it was bunching and putting pressure on my instep) and I had a rock.

The ok news: 45 F, and the 8 miles took me 1:20:05.

The bad news: this run was work…not a race effort, but work. And my goodness, I must have gotten passed scores of times. 15 years ago (or longer), I only got passed by “team” members. Then..it was the fitter looking guys passing me. Then it was the fitter looking men and women. Then it was average looking men. Now: it is average looking men and women; it is as if I am running in place. And you see the trajectory…

But there is an interesting mathematical modeling thing going on. Imagine the pace of the runners being plotted on a normal density curve; faster paces to the right, slower to the left. Also remember that, given that people start at different places on the trail, run in different directions and start at different times, you’ll neither be passed by nor pass most of the runners out there. So, I’ll have to work on this and see how one might model this. But what I do know is that I get passed by runners far more than I pass other runners. Does this mean that my pace is slower than the mean pace? My first guess is yes, but I have to think about this one.

December 28, 2013

## Tips for interpreting science results

It is a list of 20 tips for interpreting science articles that appear in the media. Many of these tips involve “how to understand statistical studies”. Here is one of the tips; the others are good too:

Regression to the mean can mislead. Extreme patterns in data are likely to be, at least in part, anomalies attributable to chance or error. The next count is likely to be less extreme. For example, if speed cameras are placed where there has been a spate of accidents, any reduction in the accident rate cannot be attributed to the camera; a reduction would probably have happened anyway.

December 8, 2013 Posted by | science, statistics | Leave a comment

## The organic foods. Don’t try them. Ever.

But wait…..well, wait nothing. There is no study which proves that organic foods don’t cause harm!

(ps: yes, I know that this is a mere correlation and no, organic foods don’t cause autism. If you don’t get the point that I am attempting to make then you aren’t the intended audience for this blog post)

November 7, 2013

## Review of Nate Silver’s book: The signal and the noise: why so may predictions fail, but some don’t

Quick Review
Excellent book. There are a few tiny technical errors (e. g., “non-linear” functions include exponential functions, but not all non-linear phenomena are exponential (e. g. power, root, logarithmic, etc.). But, aside from these, it is right on. Anyone who follows the news closely will benefit from it; I especially recommend it to those who closely follow science and politics and even sports.

It is well written and is designed for adults; it makes some (but reasonable) demands on the reader. The scientist, mathematician or engineer can read this at the end of the day but the less technically inclined will probably have to be wide awake while reading this.

Details
Silver sets you up by showing examples of failed predictions; perhaps the worst of the lot was the economic collapse in the United States prior to the 2008 general elections. Much of this was due to the collapse of the real estate market and falling house/property values. Real estate was badly overvalued, and financial firms made packages of investments whose soundness was based on many mortgages NOT defaulting at the same time; it was determined that the risk of that happening was astronomically small. That was wrong of course; one reason is that the risk of such an event is NOT described by the “normal” (bell shaped) distribution but rather by one that allows for failure with a higher degree of probability.

There were more things going on, of course; and many of these things were difficult to model accurately just due to complexity. Too many factors makes a model unusable; too few means that the model is worthless.

Silver also talks about models providing probabilistic outcomes: example saying that the GDP will be X in year Y is unrealistic; what we really should say that the probability of the GDP being X plus/minus “E” is Z percent.

Next Silver takes on pundits. In general: they don’t predict well; they are more about entertainment than anything else. Example: look at the outcome of the 2012 election; the nerds were right; the pundits (be they NPR or Fox News pundits) were wrong. NPR called the election “razor tight” (it wasn’t); Fox called it for the wrong guy. The data was clear and the sports books new this, but that doesn’t sell well, does it?

Now Silver looks at baseball. Of course there are a ton of statistics here; I am a bit sorry he didn’t introduce Bayesian analysis in this chapter though he may have been setting you up for it.

Topics include: what does raw data tell you about a player’s prospects? What role does a talent scout’s input have toward making the prediction? How does a baseball players hitting vary with age, and why is this hard to measure from the data?

The next two chapters deal with predictions: earthquakes and weather. Bottom line: we have statistical data on weather and on earthquakes, but in terms of making “tomorrow’s prediction”, we are much, much, much further along in weather than we are on earthquakes. In terms of earthquakes, we can say stuff like “region Y has a X percent chance of an earthquake of magnitude Z within the next 35 years” but that is about it. On the other hand, we are much better about, say, making forecasts of the path of a hurricane, though these are probabilistic:

In terms of weather: we have many more measurements.

But there IS the following: weather is a chaotic system; a small change in initial conditions can mean to a large change in long term outcomes. Example: one can measure a temperature at time t, but only to a certain degree of precision. The same holds for pressure, wind vectors, etc. Small perturbations can lead to very different outcomes. Solutions aren’t stable with respect to initial conditions.

You can see this easily: try to balance a pen on its tip. Physics tells us there is a precise position at which the pen is at equilibrium, even on its tip. But that equilibrium is so unstable that a small vibration of the table or even small movement of air in the room is enough to upset it.

In fact, some gambling depends on this. For example, consider a coin toss. A coin toss is governed by Newton’s laws for classical mechanics, and in principle, if you could get precise initial conditions and environmental conditions, the outcome shouldn’t be random. But it is…for practical purposes. The same holds for rolling dice.

Now what about dispensing with models and just predicting based on data alone (not regarding physical laws and relationships)? One big problem: data is noisy and is prone to be “overfitted” by a curve (or surface) that exactly matches prior data but is of no predictive value. Think of it this way: if you have n data points in the plane, there is a polynomial of degree n-1 that will fit the data EXACTLY, but in most cases have a very “wiggly” graph that provides no predictive value.

Of course that is overfitting in the extreme. Hence, most use the science of the situation to posit the type of curve that “should” provide a rough fit and then use some mathematical procedure (e. g. “least squares”) to find the “best” curve that fits.

The book goes into many more examples: example: the flu epidemic. Here one finds the old tug between models that are too simplistic to be useful for forecasting and too complicated to be used.

There are interesting sections on poker and chess and the role of probability is discussed as well as the role of machines. The poker chapter is interesting; Silver describes his experience as a poker player. He made a lot of money when poker drew lots of rookies who had money to spend; he didn’t do as well when those “bad” players left and only the most dedicated ones remained. One saw that really bad players lost more money than the best players won (not that hard to understand). He also talked about how hard it was to tell if someone was really good or merely lucky; sometimes this wasn’t perfectly clear after a few months.

Later, Silver discusses climate change and why the vast majority of scientists see it as being real and caused (or made substantially worse) by human activity. He also talks about terrorism and enemy sneak attacks; sometimes there IS a signal out there but it isn’t detected because we don’t realize that there IS a signal to detect.

However the best part of the book (and it is all pretty good, IMHO), is his discussion of Bayes law and Bayesian versus frequentist statistics. I’ve talked about this.

I’ll demonstrate Bayesian reasoning in a couple of examples, and then talk about Bayesian versus frequentist statistical testing.

Example one: back in 1999, I went to the doctor with chest pains. The doctor, based on my symptoms and my current activity level (I still swam and ran long distances with no difficulty) said it was reflux and prescribed prescription antacids. He told me this about a possible stress test: “I could stress test you but the probability of any positive being a false positive is so high, we’d learn nothing from the test”.

Example two: suppose you are testing for a drug that is not widely used; say 5 percent of the population uses it. You have a test that is 95 percent accurate in the following sense: if the person is really using the drug, it will show positive 95 percent of the time, and if the person is NOT using the drug, it will show positive only 5 percent of the time (false positive).

So now you test 2000 people for the drug. If Bob tests positive, what is the probability that he is a drug user?

Answer: There are 100 actual drug users in this population, so you’d expect 100*.95 = 95 true positives. There are 1900 non-users and 1900*.05 = 95 false positives. So there are as many false positives as true positives! The odds that someone who tests positive is really a user is 50 percent.

Now how does this apply to “hypothesis testing”?

Consider basketball. You know that a given player took 10 free shots and made 4. You wonder: what is the probability that this player is a competent free throw shooter (given competence is defined to be, say, 70 percent).

If you just go by the numbers that you see (true: n = 10 is a pathetically small sample; in real life you’d never infer anything), well, the test would be: given the probability of making a free shot is 70 percent, what is the probability that you’d see 4 (or fewer) made free shots out of 10?

Using a calculator (binomial probability calculator), we’d say there is a 4.7 percent chance we’d see 4 or fewer free shots made if the person shooting the shots was a 70 percent shooter. That is the “frequentist” way.

But suppose you found out one of the following:
1. The shooter was me (I played one season in junior high and some pick up ball many years ago…infrequently) or
2. The shooter was an NBA player.

If 1 was true, you’d believe the result or POSSIBLY say “maybe he had a good day”.
If 2 was true, then you’d say “unless this player was chosen from one of the all time worst NBA free throw shooters, he probably just had a bad day”.

Bayesian hypothesis testing gives us a way to make and informed guess. We’d ask: what is the probability that the hypothesis is true given the data that we see (asking the reverse of what the frequentist asks). But to do this, we’d have to guess: if this person is an NBA player, what is the probability, PRIOR to this 4 for 10 shooting, that this person was 70 percent or better (NBA average is about 75 percent). For the sake of argument, assume that there is a 60 percent chance that this person came from the 70 percent or better category (one could do this by seeing the percentage of NBA players shooing 70 percent of better). Assign a “bad” percentage as 50 percent (based on the worst NBA free throw shooters): (the probability of 4 or fewer made free throws out of 10 given a 50 percent free throw shooter is .377)

Then we’d use Bayes law: (.0473*.6)/(.0473*.6 + .377*.4) = .158. So it IS possible that we are seeing a decent free throw shooter having a bad day.

This has profound implications in science. For example, if one is trying to study genes versus the propensity for a given disease, there are a LOT of genes. Say one tests 1000 genes of those who had a certain type of cancer and run a study. If we accept p = .05 (5 percent) chance of having a false positive, we are likely to have 50 false positives out of this study. So, given a positive correlation between a given allele and this disease, what is the probability that this is a false positive? That is, how many true positives are we likely to have?

This is a case in which we can use the science of the situation and perhaps limit our study to genes that have some reasonable expectation of actually causing this malady. Then if we can “preassign” a probability, we might get a better feel if a positive is a false one.

Of course, this technique might induce a “user bias” into the situation from the very start.

The good news is that, given enough data, the frequentist and the Bayesian techniques converge to “the truth”.

Summary Nate Silver’s book is well written, informative and fun to read. I can recommend it without reservation.

July 22, 2013

## An ongoing application of Bayesian statistics to clinical trails

This New York Times article is about clinical trials for drugs; in particular cancer fighting drugs. The gist of the article is that there is so much human variation and so much variation between cancer cells of the “same type” of cancer is that most clinical trials prove to be disappointing.

But there is a new idea: use Bayesian statistical methods during the trial to test several drugs at once and eliminate those that appear to not be working out, even while the trial continues.

This article does NOT contain the details of how Bayesian statistics are used however.

July 16, 2013

## What do you mean by “more accurate”?

I’ve been interested in the mathematics and statistics of the breast cancer screening issue mostly because it provided a real-life application of statistics and Bayes’ Theorem.

So right now, for women between 40-49, traditional mammograms are about 80 percent accurate in the sense that, if a woman who really has breast cancer gets a mammogram, the test will catch it about 80 percent of the time. The false positive rate is about 8 percent in that: if 100 women who do NOT have breast cancer get a mammogram, 8 of the mammograms will register a “positive”.
Since the breast cancer rate for women in this age group is about 1.4 percent, there will be many more false positives than true positives; in fact a woman in this age group who gets a “positive” first mammogram has about a 16 percent chance of actually having breast cancer. I talk about these issues here.

So, suppose you desire a “more accurate test” for breast cancer. The question is this: what do you mean by “more accurate”?

1. If “more accurate” means “giving the right answer more often”, then that is pretty easy to do.
Current testing is going to be wrong: if C means cancer, N means “doesn’t have cancer”, P means “positive test” and M means “negative test”, then the probability of being wrong is:
$P(M|C)P(C) + P(P|N)P(N) = .2(.014) + .08(.986) = .08168$. On the other hand, if you just declared EVERYONE to be “cancer free”, you’d be wrong only 1.4 percent of the time! So clearly that does not work; the “false negative” rate is 100 percent, though the “false positive” rate is 0.

On the other hand if you just told everyone “you have it”, then you’d be wrong 98.6 percent of the time, but you’d have zero “false negatives”.

So being right more often isn’t what you want to maximize, and trying to minimize the false positives or the false negatives doesn’t work either.

2. So what about “detecting more of the cancer that is there”? Well, that is where this article comes in. Switching to digital mammograms does increase detection rate but also increases the number of false positives:

The authors note that for every 10,000 women 40 to 49 who are given digital mammograms, two more cases of cancer will be identified for every 170 additional false-positive examinations.

So, what one sees is that if a woman gets a positive reading, she now has an 11 percent of actually having breast cancer, though a few more cancers would be detected.

Is this progress?

My whole point: saying one test is “more accurate” than another test isn’t well defined, especially in a situation where one is trying to detect something that is relatively rare.
Here is one way to look at it: let the probability of breast cancer be $a$, the probability of detection of a cancer be given by $x$ and the probability of a false positive be given by $y$. Then the probability of a person actually having breast cancer, given a positive test is given by:
$B(x,y) =\frac{ax}{ax + (1-a)y}$; this gives us something to optimize. The partial derivatives are:
$\frac{\partial B}{\partial x}= \frac{(a)(1-a)y}{(ax+ (1-a)y)^2},\frac{\partial B}{\partial y}=\frac{(-a)(1-a)x}{(ax+ (1-a)y)^2}$. Note that $1-a$ is positive since $a$ is less than 1 (in fact, it is small). We also know that the critical point $x = y =0$ is a bit of a “duh”: find a single test that gives no false positives and no false negatives. This also shows us that our predictions will be better if $y$ goes down (fewer false positives) and if $x$ goes up (fewer false negatives). None of that is a surprise.

But of interest is in the amount of change. The denominators of each partial derivative are identical. The coefficients of the numerators are of the same magnitude; there are different signs. So the rate of improvement of the predictive value is dependent on the relative magnitudes of $x$, which is $.8$ for us, and $y$, which is $.08$. Note that $x$ is much larger than $y$ and $x$ occurs in the numerator $\frac{\partial B}{\partial y}$. Hence an increase in the accuracy of the $y$ factor (a decrease in the false positive rate) will have a greater effect on the accuracy of the test than a similar increase in the “false negative” accuracy.
Using the concept of differentials, we expect a change $\Delta x = .01$ leads to an improvement of about .00136 (substitute $x = .8, y = .08$ into the expression for $\frac{\partial B}{\partial x}$ and multiply by $.01$. Similarly an improvement (decrease) of $\Delta y = -.01$ leads to an improvement of .013609.

You can “verify” this by playing with some numbers:

Current ($x = .8, y = .08$) we get $B = .1243$. Now let’s change: $x = .81, y = .08$ leads to $B = .125693$
Now change: $x = .8, y = .07$ we get $B = .139616$

Bottom line: the best way to increase the predictive value of the test is to reduce the number of false positives, while staying the same (or improving) the percentage of “false negatives”. As things sit, the false positive rate is the bigger factor affecting predictive value.

July 12, 2013