Here is the data on alcohol consumption before and after the breakup. There is an assumption that the drinking habit increases post-breakup. Is that true?
Before
After
470
408
354
439
496
321
351
437
349
335
449
344
378
318
359
492
469
531
329
417
389
358
497
391
493
398
268
394
445
508
287
399
338
345
271
341
412
326
335
467
The null hypothesis, H0: (consumption after – before) = 0. The alternative hypothesis, HA: (consumption after – before) > 0.
T-Test
D = Mean difference of the parameter after and before mud = hypothesised mean difference Sd = Standard deviation of the difference n = number of samples
We insert the data in the following command and run the function, t.test.
Paired t-test
data: Before and After
t = -0.53754, df = 19, p-value = 0.7014
alternative hypothesis: true mean difference is greater than 0
95 percent confidence interval:
-48.49262 Inf
sample estimates:
mean difference
-11.5
There was a difference of -11.5, yet the p-value (0.7014) is higher than the critical value we chose (0.05). The test shows no evidence supporting the hypothesis.
A placebo is used in clinical trials as a control in drug studies to test the effectiveness of treatments. In drug trials, one group of participants receives the medicines (to be tested), and the other gets the placebo (say, sugar pills).
The concept of placebo stems from the assumption that a treatment has two components, one related to the specific effects of the treatment and the other, nonspecific, related to its perception. When the nonspecific effects are beneficial to the participant, it is called a placebo, and when they are harmful, it is a nocebo.
Hypothesis Testing
The null hypothesis (H0) typically represents the default state or the state of “no effect“. For example, you compare the means of two groups, such as people who took a particular drug and people who received the placebo. As a drug researcher, your objective is to find the effectiveness of the medicine. And that lays the foundation for your alternative hypothesis (HA or H1) – that the drug has a non-zero effect. The default state (H0) assumes the drug has no impact. To be specific, H0 assumes the difference between two means equals zero.
Decision trees are logical pathways created by tracing answers to multiple stages of true/false, no-go/no-go questions. Here is a representation of a simple tree based on a person liking something given their age.
Another to represent the same info is:
A regression tree is a type of decision tree. In the regression tree, each leaf represents a numeric value. The other type is a classification tree with true/false or another category in its leaves.
The power of a statistical test is the probability of rejecting the null hypothesis when the null hypothesis is false (or the effect is present). It is the right decision, and before we go deeper into it, let’s recap the two types of errors in hypothesis testing.
A type I error is when the Null hypothesis is true, but you rejected it.
A type II error is when you fail to reject a false Null hypothesis; in other words, the effect is present.
Amelie has developed a new drug against the flu and wanted to test its effectiveness. She recruited 50 people and divided them into two groups. She gave the medicine to the first group (34 people) and provided a placebo to the second (16). Here are the results.
In the treatment group, 15 out of 34 (44%) recovered in 1 day. In the control group, 4 out of 16 (25%) recovered in 1 day.
Amelie thinks her medicine is effective because of the big defence (44 – 25 = 19%) in performance and the larger sample size for the treatment group. Do you agree?
We must do a statistical test to conclude. The data is:
Sample
Events
Trials
Treatment
15
34
Control
4
16
The Null Hypothesis, H0: No impact of medicine; recovery proportion on drug equals that of placebo. The Alternative Hypothesis, H1: Medicine improves the condition; recovery proportion on drug greater than placebo.
We use the ‘prop.test()’ function in R.
prop.test(x = c(15, 4), n = c(34, 16), alternative = "greater")
We used the one-tailed test to determine whether the treatment has improved the condition compared to the control.
2-sample test for equality of proportions with continuity correction
data: c(15, 4) out of c(34, 16)
X-squared = 0.97389, df = 1, p-value = 0.1619
alternative hypothesis: greater
95 percent confidence interval:
-0.0813273 1.0000000
sample estimates:
prop 1 prop 2
0.4411765 0.2500000
The p-value of 0.1619 (more than the significance value of 0.05) is not enough to reject the null hypothesis.
What would have happened if she had recruited more people in the placebo group and found results at the same proportions?
Sample
Events
Trials
Treatment
15
34
Control
16
64
prop.test(x = c(15, 16), n = c(34, 64), alternative = "greater")
2-sample test for equality of proportions with continuity correction
data: c(15, 16) out of c(34, 64)
X-squared = 2.9205, df = 1, p-value = 0.04373
alternative hypothesis: greater
95 percent confidence interval:
0.002691967 1.000000000
sample estimates:
prop 1 prop 2
0.4411765 0.2500000
Here, the results are significant (p-value = 0.04373 < 0.05) to reject the null hypothesis in favour of the alternate (that the drug works).
What about recruiting more to the treatment group?
Sample
Events
Trials
Treatment
150
340
Control
4
16
prop.test(x = c(150, 4), n = c(340, 16), alternative = "greater")
2-sample test for equality of proportions with continuity correction
data: c(150, 4) out of c(340, 16)
X-squared = 1.5631, df = 1, p-value = 0.1056
alternative hypothesis: greater
95 percent confidence interval:
-0.02503096 1.00000000
sample estimates:
prop 1 prop 2
0.4411765 0.2500000
p-value = 0.1056; the null hypothesis stays!
Or, had many individuals in both groups?
Sample
Events
Trials
Treatment
150
340
Control
40
160
prop.test(x = c(150, 40), n = c(340, 160), alternative = "greater")
2-sample test for equality of proportions with continuity correction
data: c(150, 40) out of c(340, 160)
X-squared = 16.076, df = 1, p-value = 3.042e-05
alternative hypothesis: greater
95 percent confidence interval:
0.1149402 1.0000000
sample estimates:
prop 1 prop 2
0.4411765 0.2500000
Both the groups have plenty of samples, and now the difference (44 – 25 = 22%) overwhelmingly supports the impact of the medicine.
In the previous post, we saw how the margin of error at a specified confidence interval is estimated. The margin of error, and thereby the confidence in the data, varies with the number of samples and the confidence interval.
Here are five calculations with varying numbers of samples (500, 1041 and 2000) for three confidence intervals (90%, 95%, 99%).
As the sample size increases, the margin of error decreases, i.e., the more people you survey, the more confident you can be that your results are closer to the “true” population value (provided the sample is representative). However, the reduction diminishes as the sample size goes beyond 1000.
A recent study suggests that 63% of the people in a city believe in parapsychological phenomena. The study surveyed 1041 residents. What is the margin of error of the results at a 95% confidence interval?
This is the problem of estimating the margin of error on point estimates from a sample. You may know by now that the ultimate goal is to calculate the population proportion, for which sampling (and thus obtaining the sample proportion) is the only practical path. The point estimate of proportion, p, is evaluated from x, the number of successes (people who said “YES” to parapsychology here), and the sample size (n): p = x/n
The remaining, 1 – p, is the proportion of failures (n – x out of n).
In the given sample, 0.63 is the p or the proportion of people who believed in parapsychology. Can you conclude that 0.63 remains the fraction of similar believers in the city? The answer is No; therefore, we estimate the margin of error by applying the formula. The margin of error gives the expected range of values to capture the population at some confidence level.
margin of error = z-critical value x square root (p x (1-p) / n)
z-critical value for a 95% confidence interval is 1.96, for 99% is 2.576, etc.
The whole process can be done in one step using R.
prop.test(proportion*n_sample, n_sample, p = NULL, alternative = "two.sided",
correct = TRUE, conf.level = 0.95)
1-sample proportions test with continuity correction
data: proportion * n_sample out of n_sample, null probability 0.5
X-squared = 69.853, df = 1, p-value < 2.2e-16
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
0.5997570 0.6592715
sample estimates:
p
0.63
The margin of error is 0.029, and the confidence interval is found by adding and subtracting it from the sample proportion (0.63).
[ 0.600, 0.659]
So, we are 95% confident that the true proportion of the population lies between 0.600 and 0.659, right? Not really; it only means that if you perform several random samples from this population, we expect about 95% of those intervals to contain the true proportion.
The anthropic principle says that in the universe, our presence as observers compels conditions for our presence. While the idea was not entirely new, the phrase came up in 1973 by Brandon Carter, who proposed the weak and strong forms of anthropic principle.
The weak anthropic principle (WAP) merely says that you need to take into account those things in the environment that you can see vs those things you are unable to see. Or it simply says that the life-free universes cannot be observed it’s just a selection bias.
An example is the cosmological constant (lambda). The number must be within certain limits. If the constant were large-negative, the universe would have collapsed long ago; if it were large-positive, it would have expanded too fast for the stars to form. Either way, we wouldn’t be here to talk about it.
Amanda, Becky and Carol are at a betting station near the coin-flipping game. They observe five tosses and see all of them landing on heads.
Amanda: “It landed five times on heads; a tail is due, and I will bet on tails this time. Becky: “It is a bised coin. The probability for the next flip to land on heads is high; I will bet on heads. Carol: “Amanda has committed a fallacy. Becky may be right, but induction based on the first five tosses can still be logically incorrect. So there is no point betting either way”.
Who is right here?
Amanda has committed the Gambler’s fallacy. By expecting a tails due, she forgets about the independence of the trials. Becky’s stand is based on her interpretation of the observations. Her argument is still not logically water-tight. From the casino’s point of view, using a biased coin is risky; people like Becky will find it easily and become rich. In the absence of strong evidence, Amanda’s logic is more acceptable.
