Statistical errors and approximation27th January 2021
In statistical hypothesis testing, a type I error is the incorrect rejection of a true null hypothesis (also known as a “false positive” finding), while a type II error is incorrectly retaining a false null hypothesis (also known as a “false negative” finding). More simply stated, a type I error is to falsely infer the existence of something that is not there, while a type II error is to falsely infer the absence of something that is.
A type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis. Usually, a type I error leads one to conclude that a supposed effect or relationship exists when in fact it doesn’t. Examples of type I errors include a test that shows a patient to have a disease when in fact the patient does not have the disease, a fire alarm going on indicating a fire when in fact there is no fire, or an experiment indicating that a medical treatment should cure a disease when in fact it does not.
A type II error (or error of the second kind) is the failure to reject a false null hypothesis. Examples of type II errors would be a blood test failing to detect the disease it was designed to detect, in a patient who really has the disease; a fire breaking out and the fire alarm does not ring; or a clinical trial of a medical treatment failing to show that the treatment works when really it does.
When comparing two means, concluding the means were different when in reality they were not different would be a Type I error; concluding the means were not different when in reality they were different would be a Type II error. Various extensions have been suggested as “Type III errors”, though none have wide use.
All statistical hypothesis tests have a probability of making type I and type II errors. For example, all blood tests for a disease will falsely detect the disease in some proportion of people who don’t have it, and will fail to detect the disease in some proportion of people who do have it. A test’s probability of making a type I error is denoted by α. A test’s probability of making a type II error is denoted by β. These error rates are traded off against each other: for any given sample set, the effort to reduce one type of error generally results in increasing the other type of error. For a given test, the only way to reduce both error rates is to increase the sample size, and this may not be feasible.
Type I error
A type I error occurs when the null hypothesis (H0) is true, but is rejected. It is asserting something that is absent, a false hit. A type I error may be likened to a so-called false positive (a result that indicates that a given condition is present when it actually is not present).
In terms of folk tales, an investigator may see the wolf when there is none (“raising a false alarm”). Where the null hypothesis, H0, is: no wolf.
The type I error rate or significance level is the probability of rejecting the null hypothesis given that it is true. It is denoted by the Greek letter α (alpha) and is also called the alpha level. Often, the significance level is set to 0.05 (5%), implying that it is acceptable to have a 5% probability of incorrectly rejecting the null hypothesis.
Type II error
A type II error occurs when the null hypothesis is false, but erroneously fails to be rejected. It is failing to assert what is present, a miss. A type II error may be compared with a so-called false negative (where an actual ‘hit’ was disregarded by the test and seen as a ‘miss’) in a test checking for a single condition with a definitive result of true or false. A Type II error is committed when we fail to believe a true alternative hypothesis.
In terms of folk tales, an investigator may fail to see the wolf when it is present (“failing to raise an alarm”). Again, H0: no wolf.
The rate of the type II error is denoted by the Greek letter β (beta) and related to the power of a test (which equals 1−β).
|Table of error types||
Null hypothesis (H0) is
(probability = 1−α)
|Type II error
(probability = β)
|Reject||Type I error
(probability = α)
(probability = 1−β)
A perfect test would have zero false positives and zero false negatives. However, statistical methods are probabilistic, and it cannot be known for certain whether statistical conclusions are correct. Whenever there is uncertainty, there is the possibility of making an error. Considering this nature of statistics science, all statistical hypothesis tests have a probability of making type I and type II errors.
- The type I error rate or significance level is the probability of rejecting the null hypothesis given that it is true. It is denoted by the Greek letter α (alpha) and is also called the alpha level. Usually, the significance level is set to 0.05 (5%), implying that it is acceptable to have a 5% probability of incorrectly rejecting the true null hypothesis.
- The rate of the type II error is denoted by the Greek letter β (beta) and related to the power of a test, which equals 1−β.
These two types of error rates are traded off against each other: for any given sample set, the effort to reduce one type of error generally results in increasing the other type of error.
The quality of hypothesis test
The same idea can be expressed in terms of the rate of correct results and therefore used to minimize error rates and improve the quality of hypothesis test. To reduce the probability of committing a Type I error, making the alpha (p) value more stringent is quite simple and efficient. To decrease the probability of committing a Type II error, which is closely associated with analyses’ power, either increasing the test’s sample size or relaxing the alpha level could increase the analyses’ power. A test statistic is robust if the Type I error rate is controlled.
Varying different threshold (cut-off) value could also be used to make the test either more specific or more sensitive, which in turn elevates the test quality. For example, imagine a medical test, in which experimenter might measure the concentration of a certain protein in the blood sample. Experimenter could adjust the threshold (black vertical line in the figure) and people would be diagnosed as having diseases if any number is detected above this certain threshold. According to the image, changing the threshold would result in changes in false positives and false negatives, corresponding to movement on the curve.
Too many results are only approximate; meaning they are similar but not equal to the actual result. An approximation can turn a complex calculation into a less complicated one.
For instance, the calculation of a Poisson distribution is more complicated than that of a binomial distribution. If both only differ slightly in their end result, it is permissible to approximate the Poisson distribution by a more simple-to-use binomial distribution. Prerequisite for such approximations is a sufficient sample size. In this example, at least 100 respondents are necessary in order to justify a sufficient proximity of the two distributions. An approximation based on too small a sample can lead to errors, for example, an accidental similarity of the two distributions.
The binomial distribution can be used to solve problems such as, “If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?” The probability of exactly x heads out of N
Flips is computed using the formula:
is the number of heads (60), N is the number of flips (100), and π
is the probability of a head (0.5). Therefore, to solve this problem, you compute the probability of 60 heads, then the probability of 61 heads, 62 heads, etc, and add up all these probabilities.
Abraham de Moivre, an 18th century statistician and consultant to gamblers, was often called upon to make these lengthy computations. de Moivre noted that when the number of events (coin flips) increased, the shape of the binomial distribution approached a very smooth curve. Therefore, de Moivre reasoned that if he could find a mathematical expression for this curve, he would be able to solve problems such as finding the probability of 60 or more heads out of 100 coin flips much more easily. This is exactly what he did, and the curve he discovered is now called the normal curve. The process of using this curve to estimate the shape of the binomial distribution is known as normal approximation.
The Scope of the Normal Approximation
The scope of the normal approximation is dependent upon our sample size, becoming more accurate as the sample size grows.
The tool of normal approximation allows us to approximate the probabilities of random variables for which we don’t know all of the values, or for a very large range of potential values that would be very difficult and time consuming to calculate. We do this by converting the range of values into standardized units and finding the area under the normal curve. A problem arises when there are a limited number of samples, or draws in the case of data “drawn from a box.” A probability histogram of such a set may not resemble the normal curve, and therefore the normal curve will not accurately represent the expected values of the random variables. In other words, the scope of the normal approximation is dependent upon our sample size, becoming more accurate as the sample size grows. This characteristic follows with the statistical themes of the law of large numbers and central limit theorem.
Sixty two percent of 12th graders attend school in a particular urban school district. If a sample of 500 12th grade children are selected, find the probability that at least 290 are actually enrolled in school.
Part 1: Making the Calculations
Step 1: Find p,q, and n:
- The probability p is given in the question as 62%, or 0.62
- To find q, subtract p from 1: 1 – 0.62 = 0.38
- The sample size n is given in the question as 500
Step 2: Figure out if you can use the normal approximation to the binomial. If n * p and n * q are greater than 5, then you can use the approximation:
n * p = 310 and n * q = 190.
These are both larger than 5, so you can use the normal approximation to the binomial for this question.
Step 3: Find the mean, μ by multiplying n and p:
n * p = 310
(You actually figured that out in Step 2!).
Step 4: Multiply step 3 by q :
310 * 0.38 = 117.8.
Step 5: Take the square root of step 4 to get the standard deviation, σ:
Note: The formula for the standard deviation for a binomial is √(n*p*q).
Part 2: Using the Continuity Correction Factor
Step 6: Write the problem using correct notation. The question stated that we need to “find the probability that at least 290 are actually enrolled in school”. So:
P(X ≥ 290)
Step 7: Rewrite the problem using the continuity correction factor:
P (X ≥ 290-0.5) = P (X ≥ 289.5)
Step 8: Draw a diagram with the mean in the center. Shade the area that corresponds to the probability you are looking for. We’re looking for X ≥ 289.5, so:
Step 9: Find the z-score.
You can find this by subtracting the mean (μ) from the probability you found in step 7, then dividing by the standard deviation (σ):
(289.5 – 310) / 10.85 = -1.89
Step 10: Look up the z-value in the z-table:
The area for -1.89 is 0.4706.
Step 11: Add .5 to your answer in step 10 to find the total area pictured:
0.4706 + 0.5 = 0.9706.
That’s it! The probability is .9706, or 97.06%.
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