hypothesis test beta

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What is a Beta Level in Statistics? (Definition & Example)

In statistics, we use hypothesis tests to determine if some assumption about a population parameter is true.

A hypothesis test always has the following two hypotheses:

Null hypothesis (H 0 ): The sample data is consistent with the prevailing belief about the population parameter.

Alternative hypothesis (H A ): The sample data suggests that the assumption made in the null hypothesis is not true. In other words, there is some non-random cause influencing the data.

Whenever we conduct a hypothesis test, there are always four possible outcomes:

There are two types of errors we can commit:

• Type I Error: We reject the null hypothesis when it is actually true. The probability of committing this type of error is denoted as α .
• Type II Error: We fail to reject the null hypothesis when it is actually false. The probability of committing this type of error is denoted as β .

The Relationship Between Alpha and Beta

Ideally researchers want both the probability of committing a type I error and the probability of committing a type II error to be low.

However, a tradeoff exists between these two probabilities. If we decrease the alpha level, we can decrease the probability of rejecting a null hypothesis when it’s actually true, but this actually increases the beta level – the probability that we fail to reject the null hypothesis when it actually is false.

The Relationship Between Power and Beta

The power of a hypothesis test refers to the probability of detecting an effect or difference when an effect or difference is actually present. In other words, it’s the probability of correctly rejecting a false null hypothesis.

It is calculated as:

Power = 1 – β

In general, researchers want the power of a test to be high so that if some effect or difference does exist, the test is able to detect it.

From the equation above, we can see that the best way to raise the power of a test is to reduce the beta level. And the best way to reduce the beta level is typically to increase the sample size.

The following examples shows how to calculate the beta level of a hypothesis test and demonstrate why increasing the sample size can lower the beta level.

Example 1: Calculate Beta for a Hypothesis Test

Suppose a researcher wants to test if the mean weight of widgets produced at a factory is less than 500 ounces. It is known that the standard deviation of the weights is 24 ounces and the researcher decides to collect a random sample of 40 widgets.

He will perform the following hypothesis at α = 0.05:

• H 0 : μ = 500
• H A : μ < 500

Now imagine that the mean weight of widgets being produced is actually 490 ounces. In other words, the null hypothesis should be rejected.

We can use the following steps to calculate the beta level – the probability of failing to reject the null hypothesis when it actually should be rejected:

Step 1: Find the non-rejection region.

According to the Critical Z Value Calculator , the left-tailed critical value at α = 0.05 is -1.645 .

Step 2: Find the minimum sample mean we will fail to reject.

The test statistic is calculated as z = ( x – μ) / (s/√ n )

Thus, we can solve this equation for the sample mean:

• x = μ – z*(s/√ n )
• x = 500 – 1.645*(24/√ 40 )
• x = 493.758

Step 3: Find the probability of the minimum sample mean actually occurring.

We can calculate this probability as:

• P(Z ≥ (493.758 – 490) / (24/√ 40 ))
• P(Z ≥ 0.99)

According to the Normal CDF Calculator , the probability that Z ≥ 0.99 is  0.1611 .

Thus, the beta level for this test is β = 0.1611. This means there is a 16.11% chance of failing to detect the difference if the real mean is 490 ounces.

Example 2: Calculate Beta for a Test with a Larger Sample Size

Now suppose the researcher performs the exact same hypothesis test but instead uses a sample size of n = 100 widgets. We can repeat the same three steps to calculate the beta level for this test:

• x = 500 – 1.645*(24/√ 100 )
• P(Z ≥ (496.05 – 490) / (24/√ 100 ))
• P(Z ≥ 2.52)

According to the Normal CDF Calculator , the probability that Z ≥ 2.52 is 0.0059.

Thus, the beta level for this test is β = 0.0059. This means there is only a .59% chance of failing to detect the difference if the real mean is 490 ounces.

Notice that by simply increasing the sample size from 40 to 100, the researcher was able to reduce the beta level from 0.1611 all the way down to .0059.

Bonus: Use this Type II Error Calculator to automatically calculate the beta level of a test.

Additional Resources

Introduction to Hypothesis Testing How to Write a Null Hypothesis (5 Examples) An Explanation of P-Values and Statistical Significance

Featured Posts

Hey there. My name is Zach Bobbitt. I have a Master of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

One Reply to “What is a Beta Level in Statistics? (Definition & Example)”

Your site is really great! Many things are explained clearly and to the point! Very helpful!

Nevertheless, as I’m not an expert I have difficulty in understanding why alpha and beta are not equal. Why does decreasing the alpha-level increase the beta-level? Is there an explanation why and how alpha and beta are connected to each other?

Thanks a lot in advance! Kind regards Peter Kleindienst

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In hypothesis testing, there are two important values you should be familiar with: alpha (α) and beta (β). These values are used to determine how meaningful the results of the test are. So, let’s talk about them!

Alpha is also known as the level of significance. This represents the probability of obtaining your results due to chance. The smaller this value is, the more “unusual” the results, indicating that the sample is from a different population than it’s being compared to, for example. Commonly, this value is set to .05 (or 5%), but can take on any value chosen by the research not exceeding .05.

Alpha also represents your chance of making a Type I Error . What’s that? The chance that you reject the null hypothesis when in reality, you should fail to reject the null hypothesis. In other words, your sample data indicates that there is a difference when in reality, there is not. Like a false positive.

Multiple Hypothesis Testing

When a study includes more than one hypothesis test, the alpha of the test will not match the alpha for each test. There is a cumulative effect of alpha when multiple tests are being conducted such that three tests using alpha=.05 each would have a cumulative alpha of .15 for the study. This exceeds what is acceptable for quantitative research. Therefore, researchers should consider making an adjustment, such as a  Bonferroni Correction . Using this method, the researcher takes the alpha of the study and divides it by the number of tests being conducted: .05/5 = .01. The result is the level of significance that will be used for each test to determine significance.

The other key-value relates to the power of your study. Power refers to your study’s ability to find a difference if there is one. It logically follows that the greater the power, the more meaningful your results are. Beta = 1 – Power. Values of beta should be kept small, but do not have to be as small as alpha values. Values between .05 and .20 are acceptable.

Beta also represents the chance of making a Type II Error . As you may have guessed, this means that you came to the wrong conclusion in your study, but it’s the opposite of a Type I Error. With a Type II Error, you incorrectly fail to reject the null. In simpler terms, the data indicates that there is not a significant difference when in reality there is. Your study failed to capture a significant finding. Like a false negative.

Type I Error: Testing positive for antibodies, when in fact, no antibodies are present. Type II Error: Testing negative for antibodies when in fact, antibodies are present.

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A Guide on Data Analysis

14 hypothesis testing.

Error types:

Type I Error (False Positive):

• Reality: nope
• Diagnosis/Analysis: yes

Type II Error (False Negative):

• Reality: yes
• Diagnosis/Analysis: nope

Power: The probability of rejecting the null hypothesis when it is actually false

Always written in terms of the population parameter ( \(\beta\) ) not the estimator/estimate ( \(\hat{\beta}\) )

Sometimes, different disciplines prefer to use \(\beta\) (i.e., standardized coefficient), or \(\mathbf{b}\) (i.e., unstandardized coefficient)

\(\beta\) and \(\mathbf{b}\) are similar in interpretation; however, \(\beta\) is scale free. Hence, you can see the relative contribution of \(\beta\) to the dependent variable. On the other hand, \(\mathbf{b}\) can be more easily used in policy decisions.

\[ \beta_j = \mathbf{b} \frac{s_{x_j}}{s_y} \]

Assuming the null hypothesis is true, what is the (asymptotic) distribution of the estimator

\[ \begin{aligned} &H_0: \beta_j = 0 \\ &H_1: \beta_j \neq 0 \end{aligned} \]

then under the null, the OLS estimator has the following distribution

\[ A1-A3a, A5: \sqrt{n} \hat{\beta_j} \sim N(0,Avar(\sqrt{n}\hat{\beta}_j)) \]

• For the one-sided test, the null is a set of values, so now you choose the worst case single value that is hardest to prove and derive the distribution under the null

\[ \begin{aligned} &H_0: \beta_j\ge 0 \\ &H_1: \beta_j < 0 \end{aligned} \]

then the hardest null value to prove is \(H_0: \beta_j=0\) . Then under this specific null, the OLS estimator has the following asymptotic distribution

\[ A1-A3a, A5: \sqrt{n}\hat{\beta_j} \sim N(0,Avar(\sqrt{n}\hat{\beta}_j)) \]

14.1 Types of hypothesis testing

\(H_0 : \theta = \theta_0\)

\(H_1 : \theta \neq \theta_0\)

How far away / extreme \(\theta\) can be if our null hypothesis is true

Assume that our likelihood function for q is \(L(q) = q^{30}(1-q)^{70}\) Likelihood function

Log-Likelihood function

Figure from ( Fox 1997 )

typically, The likelihood ratio test (and Lagrange Multiplier (Score) ) performs better with small to moderate sample sizes, but the Wald test only requires one maximization (under the full model).

14.2 Wald test

\[ \begin{aligned} W &= (\hat{\theta}-\theta_0)'[cov(\hat{\theta})]^{-1}(\hat{\theta}-\theta_0) \\ W &\sim \chi_q^2 \end{aligned} \]

where \(cov(\hat{\theta})\) is given by the inverse Fisher Information matrix evaluated at \(\hat{\theta}\) and q is the rank of \(cov(\hat{\theta})\) , which is the number of non-redundant parameters in \(\theta\)

Alternatively,

\[ t_W=\frac{(\hat{\theta}-\theta_0)^2}{I(\theta_0)^{-1}} \sim \chi^2_{(v)} \]

where v is the degree of freedom.

Equivalently,

\[ s_W= \frac{\hat{\theta}-\theta_0}{\sqrt{I(\hat{\theta})^{-1}}} \sim Z \]

How far away in the distribution your sample estimate is from the hypothesized population parameter.

For a null value, what is the probability you would have obtained a realization “more extreme” or “worse” than the estimate you actually obtained?

Significance Level ( \(\alpha\) ) and Confidence Level ( \(1-\alpha\) )

• The significance level is the benchmark in which the probability is so low that we would have to reject the null
• The confidence level is the probability that sets the bounds on how far away the realization of the estimator would have to be to reject the null.

Test Statistics

• Standardized (transform) the estimator and null value to a test statistic that always has the same distribution
• Test Statistic for the OLS estimator for a single hypothesis

\[ T = \frac{\sqrt{n}(\hat{\beta}_j-\beta_{j0})}{\sqrt{n}SE(\hat{\beta_j})} \sim^a N(0,1) \]

\[ T = \frac{(\hat{\beta}_j-\beta_{j0})}{SE(\hat{\beta_j})} \sim^a N(0,1) \]

the test statistic is another random variable that is a function of the data and null hypothesis.

• T denotes the random variable test statistic
• t denotes the single realization of the test statistic

Evaluating Test Statistic: determine whether or not we reject or fail to reject the null hypothesis at a given significance / confidence level

Three equivalent ways

Critical Value

• Confidence Interval

For a given significance level, will determine the critical value \((c)\)

• One-sided: \(H_0: \beta_j \ge \beta_{j0}\)

\[ P(T<c|H_0)=\alpha \]

Reject the null if \(t<c\)

• One-sided: \(H_0: \beta_j \le \beta_{j0}\)

\[ P(T>c|H_0)=\alpha \]

Reject the null if \(t>c\)

• Two-sided: \(H_0: \beta_j \neq \beta_{j0}\)

\[ P(|T|>c|H_0)=\alpha \]

Reject the null if \(|t|>c\)

Calculate the probability that the test statistic was worse than the realization you have

\[ \text{p-value} = P(T<t|H_0) \]

\[ \text{p-value} = P(T>t|H_0) \]

\[ \text{p-value} = P(|T|<t|H_0) \]

reject the null if p-value \(< \alpha\)

Using the critical value associated with a null hypothesis and significance level, create an interval

\[ CI(\hat{\beta}_j)_{\alpha} = [\hat{\beta}_j-(c \times SE(\hat{\beta}_j)),\hat{\beta}_j+(c \times SE(\hat{\beta}_j))] \]

If the null set lies outside the interval then we reject the null.

• We are not testing whether the true population value is close to the estimate, we are testing that given a field true population value of the parameter, how like it is that we observed this estimate.
• Can be interpreted as we believe with \((1-\alpha)\times 100 \%\) probability that the confidence interval captures the true parameter value.

With stronger assumption (A1-A6), we could consider Finite Sample Properties

\[ T = \frac{\hat{\beta}_j-\beta_{j0}}{SE(\hat{\beta}_j)} \sim T(n-k) \]

• This above distributional derivation is strongly dependent on A4 and A5
• T has a student t-distribution because the numerator is normal and the denominator is \(\chi^2\) .
• Critical value and p-values will be calculated from the student t-distribution rather than the standard normal distribution.
• \(n \to \infty\) , \(T(n-k)\) is asymptotically standard normal.

Rule of thumb

if \(n-k>120\) : the critical values and p-values from the t-distribution are (almost) the same as the critical values and p-values from the standard normal distribution.

if \(n-k<120\)

• if (A1-A6) hold then the t-test is an exact finite distribution test
• if (A1-A3a, A5) hold, because the t-distribution is asymptotically normal, computing the critical values from a t-distribution is still a valid asymptotic test (i.e., not quite the right critical values and p0values, the difference goes away as \(n \to \infty\) )

14.2.1 Multiple Hypothesis

test multiple parameters as the same time

• \(H_0: \beta_1 = 0\ \& \ \beta_2 = 0\)
• \(H_0: \beta_1 = 1\ \& \ \beta_2 = 0\)

perform a series of simply hypothesis does not answer the question (joint distribution vs. two marginal distributions).

The test statistic is based on a restriction written in matrix form.

\[ y=\beta_0+x_1\beta_1 + x_2\beta_2 + x_3\beta_3 + \epsilon \]

Null hypothesis is \(H_0: \beta_1 = 0\) & \(\beta_2=0\) can be rewritten as \(H_0: \mathbf{R}\beta -\mathbf{q}=0\) where

• \(\mathbf{R}\) is a \(m \times k\) matrix where m is the number of restrictions and \(k\) is the number of parameters. \(\mathbf{q}\) is a \(k \times 1\) vector
• \(\mathbf{R}\) “picks up” the relevant parameters while \(\mathbf{q}\) is a the null value of the parameter

\[ \mathbf{R}= \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} \right), \mathbf{q} = \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \]

Test Statistic for OLS estimator for a multiple hypothesis

\[ F = \frac{(\mathbf{R\hat{\beta}-q})\hat{\Sigma}^{-1}(\mathbf{R\hat{\beta}-q})}{m} \sim^a F(m,n-k) \]

\(\hat{\Sigma}^{-1}\) is the estimator for the asymptotic variance-covariance matrix

• if A4 holds, both the homoskedastic and heteroskedastic versions produce valid estimator
• If A4 does not hold, only the heteroskedastic version produces valid estimators.

When \(m = 1\) , there is only a single restriction, then the \(F\) -statistic is the \(t\) -statistic squared.

\(F\) distribution is strictly positive, check F-Distribution for more details.

14.2.2 Linear Combination

Testing multiple parameters as the same time

\[ \begin{aligned} H_0&: \beta_1 -\beta_2 = 0 \\ H_0&: \beta_1 - \beta_2 > 0 \\ H_0&: \beta_1 - 2\times\beta_2 =0 \end{aligned} \]

Each is a single restriction on a function of the parameters.

Null hypothesis:

\[ H_0: \beta_1 -\beta_2 = 0 \]

can be rewritten as

\[ H_0: \mathbf{R}\beta -\mathbf{q}=0 \]

where \(\mathbf{R}\) =(0 1 -1 0 0) and \(\mathbf{q}=0\)

14.2.3 Estimate Difference in Coefficients

There is no package to estimate for the difference between two coefficients and its CI, but a simple function created by Katherine Zee can be used to calculate this difference. Some modifications might be needed if you don’t use standard lm model in R.

14.2.4 Application

14.2.5 nonlinear.

Suppose that we have q nonlinear functions of the parameters \[ \mathbf{h}(\theta) = \{ h_1 (\theta), ..., h_q (\theta)\}' \]

The,n, the Jacobian matrix ( \(\mathbf{H}(\theta)\) ), of rank q is

\[ \mathbf{H}_{q \times p}(\theta) = \left( \begin{array} {ccc} \frac{\partial h_1(\theta)}{\partial \theta_1} & ... & \frac{\partial h_1(\theta)}{\partial \theta_p} \\ . & . & . \\ \frac{\partial h_q(\theta)}{\partial \theta_1} & ... & \frac{\partial h_q(\theta)}{\partial \theta_p} \end{array} \right) \]

where the null hypothesis \(H_0: \mathbf{h} (\theta) = 0\) can be tested against the 2-sided alternative with the Wald statistic

\[ W = \frac{\mathbf{h(\hat{\theta})'\{H(\hat{\theta})[F(\hat{\theta})'F(\hat{\theta})]^{-1}H(\hat{\theta})'\}^{-1}h(\hat{\theta})}}{s^2q} \sim F_{q,n-p} \]

14.3 The likelihood ratio test

\[ t_{LR} = 2[l(\hat{\theta})-l(\theta_0)] \sim \chi^2_v \]

Compare the height of the log-likelihood of the sample estimate in relation to the height of log-likelihood of the hypothesized population parameter

This test considers a ratio of two maximizations,

\[ \begin{aligned} L_r &= \text{maximized value of the likelihood under $H_0$ (the reduced model)} \\ L_f &= \text{maximized value of the likelihood under $H_0 \cup H_a$ (the full model)} \end{aligned} \]

Then, the likelihood ratio is:

\[ \Lambda = \frac{L_r}{L_f} \]

which can’t exceed 1 (since \(L_f\) is always at least as large as \(L-r\) because \(L_r\) is the result of a maximization under a restricted set of the parameter values).

The likelihood ratio statistic is:

\[ \begin{aligned} -2ln(\Lambda) &= -2ln(L_r/L_f) = -2(l_r - l_f) \\ \lim_{n \to \infty}(-2ln(\Lambda)) &\sim \chi^2_v \end{aligned} \]

where \(v\) is the number of parameters in the full model minus the number of parameters in the reduced model.

If \(L_r\) is much smaller than \(L_f\) (the likelihood ratio exceeds \(\chi_{\alpha,v}^2\) ), then we reject he reduced model and accept the full model at \(\alpha \times 100 \%\) significance level

14.4 Lagrange Multiplier (Score)

\[ t_S= \frac{S(\theta_0)^2}{I(\theta_0)} \sim \chi^2_v \]

where \(v\) is the degree of freedom.

Compare the slope of the log-likelihood of the sample estimate in relation to the slope of the log-likelihood of the hypothesized population parameter

14.5 Two One-Sided Tests (TOST) Equivalence Testing

This is a good way to test whether your population effect size is within a range of practical interest (e.g., if the effect size is 0).

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Understanding Hypothesis Tests: Significance Levels (Alpha) and P values in Statistics

Topics: Hypothesis Testing , Statistics

What do significance levels and P values mean in hypothesis tests? What is statistical significance anyway? In this post, I’ll continue to focus on concepts and graphs to help you gain a more intuitive understanding of how hypothesis tests work in statistics.

To bring it to life, I’ll add the significance level and P value to the graph in my previous post in order to perform a graphical version of the 1 sample t-test. It’s easier to understand when you can see what statistical significance truly means!

Here’s where we left off in my last post . We want to determine whether our sample mean (330.6) indicates that this year's average energy cost is significantly different from last year’s average energy cost of $260.

The probability distribution plot above shows the distribution of sample means we’d obtain under the assumption that the null hypothesis is true (population mean = 260) and we repeatedly drew a large number of random samples.

I left you with a question: where do we draw the line for statistical significance on the graph? Now we'll add in the significance level and the P value, which are the decision-making tools we'll need.

We'll use these tools to test the following hypotheses:

• Null hypothesis: The population mean equals the hypothesized mean (260).
• Alternative hypothesis: The population mean differs from the hypothesized mean (260).

What Is the Significance Level (Alpha)?

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

These types of definitions can be hard to understand because of their technical nature. A picture makes the concepts much easier to comprehend!

The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis.

In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we’d expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis.

Our sample mean (330.6) falls within the critical region, which indicates it is statistically significant at the 0.05 level.

We can also see if it is statistically significant using the other common significance level of 0.01.

The two shaded areas each have a probability of 0.005, which adds up to a total probability of 0.01. This time our sample mean does not fall within the critical region and we fail to reject the null hypothesis. This comparison shows why you need to choose your significance level before you begin your study. It protects you from choosing a significance level because it conveniently gives you significant results!

Thanks to the graph, we were able to determine that our results are statistically significant at the 0.05 level without using a P value. However, when you use the numeric output produced by statistical software , you’ll need to compare the P value to your significance level to make this determination.

Ready for a demo of Minitab Statistical Software? Just ask! 

What Are P values?

P-values are the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.

This definition of P values, while technically correct, is a bit convoluted. It’s easier to understand with a graph!

To graph the P value for our example data set, we need to determine the distance between the sample mean and the null hypothesis value (330.6 - 260 = 70.6). Next, we can graph the probability of obtaining a sample mean that is at least as extreme in both tails of the distribution (260 +/- 70.6).

In the graph above, the two shaded areas each have a probability of 0.01556, for a total probability 0.03112. This probability represents the likelihood of obtaining a sample mean that is at least as extreme as our sample mean in both tails of the distribution if the population mean is 260. That’s our P value!

When a P value is less than or equal to the significance level, you reject the null hypothesis. If we take the P value for our example and compare it to the common significance levels, it matches the previous graphical results. The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level.

If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260.

A common mistake is to interpret the P-value as the probability that the null hypothesis is true. To understand why this interpretation is incorrect, please read my blog post  How to Correctly Interpret P Values .

Discussion about Statistically Significant Results

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that we can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by:

• The assumption that the null hypothesis is true—the graphs are centered on the null hypothesis value.
• The significance level—how far out do we draw the line for the critical region?
• Our sample statistic—does it fall in the critical region?

Keep in mind that there is no magic significance level that distinguishes between the studies that have a true effect and those that don’t with 100% accuracy. The common alpha values of 0.05 and 0.01 are simply based on tradition. For a significance level of 0.05, expect to obtain sample means in the critical region 5% of the time when the null hypothesis is true . In these cases, you won’t know that the null hypothesis is true but you’ll reject it because the sample mean falls in the critical region. That’s why the significance level is also referred to as an error rate!

This type of error doesn’t imply that the experimenter did anything wrong or require any other unusual explanation. The graphs show that when the null hypothesis is true, it is possible to obtain these unusual sample means for no reason other than random sampling error. It’s just luck of the draw.

Significance levels and P values are important tools that help you quantify and control this type of error in a hypothesis test. Using these tools to decide when to reject the null hypothesis increases your chance of making the correct decision.

If you like this post, you might want to read the other posts in this series that use the same graphical framework:

• Previous: Why We Need to Use Hypothesis Tests
• Next: Confidence Intervals and Confidence Levels

If you'd like to see how I made these graphs, please read: How to Create a Graphical Version of the 1-sample t-Test .

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Confusing Statistical Terms #2: Alpha and Beta

by Karen Grace-Martin   28 Comments

Oh so many years ago I had my first insight into just how ridiculously confusing all the statistical terminology can be for novices.

I was TAing a two-semester applied statistics class for graduate students in biology.  It started with basic hypothesis testing and went on through to multiple regression.

It was a cross-listed class, meaning there were a handful of courageous (or masochistic) undergrads in the class, and they were having trouble keeping up with the ambitious graduate-level pace.

I remember one day in particular.  I was leading a discussion section when one of the poor undergrads was hopelessly lost.  We were talking about the simple regression–a regression model with only one predictor variable. She was stuck on understanding the regression coefficient (beta) and the intercept .

In most textbooks, the regression slope coefficient is denoted by β 1 and the intercept is denoted by β 0 . But in the one we were using (and I’ve seen this in others) the regression slope coefficient was denoted by β (beta), and the intercept was denoted by α (alpha). I guess the advantage of this is to not have to include subscripts.

It was only after repeated probing that I realized she was logically trying to fit what we were talking about into the concepts of alpha and beta that we had already taught her–Type I and Type II errors in hypothesis testing.

Entirely.  Different.  Concepts.

With the same names.

Once I realized the source of the misunderstanding, I was able to explain that we were using the same terminology for entirely different concepts.

But as it turns out, there are even more meanings of both alpha and beta in statistics.   Here they all are:

Hypothesis testing

As I already mentioned, the definition most learners of statistics come to first for beta and alpha are about hypothesis testing.

α (Alpha) is the probability of Type I error in any hypothesis test–incorrectly rejecting the null hypothesis.

β (Beta) is the probability of Type II error in any hypothesis test–incorrectly failing to reject the null hypothesis.  (1 – β is power).

Population Regression coefficients

In most textbooks and software packages, the population regression coefficients are denoted by β.

Like all population parameters, they are theoretical–we don’t know their true values.  The regression coefficients we estimate from our sample are estimates of those parameter values.  Most parameters are denoted with Greek letters and statistics with the corresponding Latin letters.

Most texts refer to the intercept as β 0 ( beta-naught )  and every other regression coefficient as β 1 , β 2 , β 3 , etc.  But as I already mentioned, some statistics texts will refer to the intercept as α, to distinguish it from the other coefficients.

If the β has a ^ over it, it’s called beta-hat and is the sample estimate of the population parameter β. And to make that even more confusing, sometimes instead of beta-hat, those sample estimates are denoted B or b.

Standardized Regression Coefficient Estimates

But, for some reason, SPSS labels standardized regression coefficient estimates as Beta .  Despite the fact that they are statistics–measured on the sample, not the population.

More confusion.

And I can’t verify this, but I vaguely recall that Systat uses the same term.  If you have Systat and can verify or negate this claim, feel free to do so in the comments.

Cronbach’s alpha

Another, completely separate use of alpha is Cronbach’s alpha , aka Coefficient Alpha, which measures the reliability of a scale.

It’s a very useful little statistic, but should not be confused with either of the other uses of alpha.

Beta Distribution and Beta Regression

You may have also heard of Beta regression, which is a generalized linear model based on the beta distribution .

The beta distribution is another distribution in statistics, just like the normal, Poisson, or binomial distributions. There are dozens of distributions in statistics, but some are used and taught more than others, so you may not have heard of this one.

The beta distribution has nothing to do with any of the other uses of the term beta.

Other uses of Alpha and Beta

If you really start to get into higher level statistics, you’ll see alpha and beta used quite often as parameters in different distributions. I don’t know if they’re commonly used simply because everyone knows those Greek letters. But you’ll see them, for example, as parameters of a gamma distribution. Relatedly, you’ll see alpha as a parameter of a negative binomial distribution.

If you think of other uses of alpha or beta, please leave them in the comments.

See the full Series on Confusing Statistical Terms .

Reader Interactions

April 26, 2021 at 8:49 am

The first place I think the terminology drives my high school kids crazy is when we no longer write the equation of a line as y = mx + b like in algebra, but write it y = a + bx.

In algebra, b was the y-intercept. Now it’s slope. Geez, why do we do this to kids?

June 9, 2021 at 11:21 am

To be fair, one very important lesson in ones journey through maths is to not stick too much with certain letters for variables. It is always in relation to something. A hard lesson but a worthy one, imo.

June 22, 2021 at 9:53 am

I think it’s because in statistics, we use m for means. I don’t know why all the algebra textbooks use m for a slope, but yes, I get your point.

But I’ve found that pointing it out that this is confusing is really helpful to people who are confused.

June 6, 2020 at 8:06 pm

I really like the Hypothesis Testing graph. It’s one of the most comprehensive I’ve seen. Thank you for posting!

June 30, 2018 at 11:00 am

Thank you this is very helpful. I’m new to genomics and I get confused about alpha and beta when people talk about it. In genomics people use both regression and hypothesis testing frequently so I’m getting more confused and mixing up the betas. Now it’s clear to me after reading this post. Can you please talk about effect size and p-values as well?

January 20, 2017 at 1:04 pm

kindly tell me if alpha+beta what will be answer

April 15, 2017 at 12:53 am

It’s a formulae…apha×beta =-b/a

April 15, 2017 at 12:54 am

Sorry…alpha+beta=-b/a

December 14, 2016 at 3:59 pm

Hi, I am comparing stock returns on a monthly and daily basis, there are differences between the outcomes of the Hypothesis tests. Could you tell me a possible reason/s for the results?

September 14, 2016 at 11:52 am

I have a confussion regarding the name and use of the product of dividing the estimator (coefficient) of a variable by its S.E.. In some places I found the called this Est./S.E. as standardized regression coefficient, is that right? Thanks

February 29, 2016 at 12:27 pm

hey, i was wondering if you can explain to me the assumptions that are needed for a and b to be unbiased estimates of Alpha and Beta. thanks ,

November 17, 2015 at 10:03 pm

Thank you so much! You are very kind for spending your time to help others. Bless you and your family

April 10, 2015 at 11:54 am

hi, i am very new to stats and i am doing a multiple regression analysis in spss and two letters confuse me. The spss comes up with a B letter (capital) but here i see all of you talking about β (greek small letter), and when i listen to youtube videos i hear beta wades, what is their difference? Please help!!!!

January 20, 2017 at 3:32 pm

calli, its beta weight.. its a standarized regression coefficient [slope of a line in a regression equation]. it equals correlation when there is a single predictor.

November 14, 2014 at 3:16 pm

Can you tell me why we use alpha?

July 5, 2013 at 10:51 am

wha is bifference between beta and beta hat and u and ui hat

July 8, 2013 at 2:59 pm

Hi Ayesha, great question. The terms without hats are the population parameters. The terms with hats indicate the sample statistic, which estimates the population parameter.

September 14, 2019 at 2:02 am

but the population parameters are only theoretical, because we can’t get the entire data of nature and society to research? Is that so?

October 28, 2019 at 10:15 am

July 30, 2012 at 1:46 am

Im wondering about the use of “beta 0” In a null hypothesis. What im wanting to test is “The effect of diameter on height = 0, or not equal to 0.

Having a lil trouble remembering the stat101 terminology.

I got the impression that that rather than writing: Ho: Ed on H = 0 Ha: Ed on H ≠ 0 can I use the beta nought symbol like B1 – B2 = 0 etc instead or am I way off track?

August 3, 2012 at 2:57 pm

The effect of diameter on height is most likely the slope, not the intercept. It’s beta1 in this equation:

Height=beta0 + beta1*diameter

Here’s more info about the intercept: https://www.theanalysisfactor.com/interpreting-the-intercept-in-a-regression-model/

September 29, 2011 at 5:16 am

This is so helpful. Thx!!

March 20, 2011 at 4:38 pm

I have read the Type I and Type II distinction about 20 times and still have been confused. I have created mnemonic devices, used visual imagery – the whole nine yards. I just read your description and it clicked. Easy peasy. Thanks!

March 25, 2011 at 12:54 pm

Thanks, Carrie! Glad it was helpful.

February 11, 2011 at 9:54 pm

Hi! This helps, but I am a little confused about this article I am reading. There is a table that lists the variables with Standardized Regression Coefficients. Two of the coefficients have ***. The *** has a note that says “alpha > 0.01”. What is alpha in this case? Is it the intercept? Is this note indicating that these variables are not significant because they are > 0.01? Damn statistics! Why can’t things be less confusing!?!?!

February 18, 2011 at 6:27 pm

Hi Lyndsey,

That’s pretty strange. It’s pretty common to have *** next to coefficients that are significant, i.e. p “, “not p <". And while yes, you want to compare p to alpha, that statement is no equivalent. I'd have to see it to really make sense of it. Can you give us a link?

December 18, 2009 at 8:10 am

I find SPSS’s use of beta for standardised coefficients tremendously annoying!

BTW a beta with a hat on is sometimes used to denote the sample estimate of the population parameter. But mathematicians tend to use any greek letters they feel like using! The trick for maintaining sanity is always to introduce what symbols denote.

December 21, 2009 at 5:38 pm

Ah, yes! Beta hats. This is actually “standard” statistical notation. The sample estimate of any population parameter puts a hat on the parameter. So if beta is the parameter, beta hat is the estimate of that parameter value.

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12.2.1: Hypothesis Test for Linear Regression

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• Rachel Webb
• Portland State University

To test to see if the slope is significant we will be doing a two-tailed test with hypotheses. The population least squares regression line would be \(y = \beta_{0} + \beta_{1} + \varepsilon\) where \(\beta_{0}\) (pronounced “beta-naught”) is the population \(y\)-intercept, \(\beta_{1}\) (pronounced “beta-one”) is the population slope and \(\varepsilon\) is called the error term.

If the slope were horizontal (equal to zero), the regression line would give the same \(y\)-value for every input of \(x\) and would be of no use. If there is a statistically significant linear relationship then the slope needs to be different from zero. We will only do the two-tailed test, but the same rules for hypothesis testing apply for a one-tailed test.

We will only be using the two-tailed test for a population slope.

The hypotheses are:

\(H_{0}: \beta_{1} = 0\) \(H_{1}: \beta_{1} \neq 0\)

The null hypothesis of a two-tailed test states that there is not a linear relationship between \(x\) and \(y\). The alternative hypothesis of a two-tailed test states that there is a significant linear relationship between \(x\) and \(y\).

Either a t-test or an F-test may be used to see if the slope is significantly different from zero. The population of the variable \(y\) must be normally distributed.

F-Test for Regression

An F-test can be used instead of a t-test. Both tests will yield the same results, so it is a matter of preference and what technology is available. Figure 12-12 is a template for a regression ANOVA table,

where \(n\) is the number of pairs in the sample and \(p\) is the number of predictor (independent) variables; for now this is just \(p = 1\). Use the F-distribution with degrees of freedom for regression = \(df_{R} = p\), and degrees of freedom for error = \(df_{E} = n - p - 1\). This F-test is always a right-tailed test since ANOVA is testing the variation in the regression model is larger than the variation in the error.

Use an F-test to see if there is a significant relationship between hours studied and grade on the exam. Use \(\alpha\) = 0.05.

T-Test for Regression

If the regression equation has a slope of zero, then every \(x\) value will give the same \(y\) value and the regression equation would be useless for prediction. We should perform a t-test to see if the slope is significantly different from zero before using the regression equation for prediction. The numeric value of t will be the same as the t-test for a correlation. The two test statistic formulas are algebraically equal; however, the formulas are different and we use a different parameter in the hypotheses.

The formula for the t-test statistic is \(t = \frac{b_{1}}{\sqrt{ \left(\frac{MSE}{SS_{xx}}\right) }}\)

Use the t-distribution with degrees of freedom equal to \(n - p - 1\).

The t-test for slope has the same hypotheses as the F-test:

Use a t-test to see if there is a significant relationship between hours studied and grade on the exam, use \(\alpha\) = 0.05.

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How to Find the Beta With an Alpha Hypothesis

How to Calculate a P-Value

In all statistical hypothesis tests, there are two especially important statistics -- alpha and beta. These values represent, respectively, the probability of a type I error and the probability of a type II error. A type I error is a false positive, or conclusion that states there is a significant relationship in the data when in fact there is no significant relationship. A type II error is a false negative, or conclusion that states there is no relationship in the data when in fact there is a significant relationship. Usually, beta is difficult to find. However, if you already have an alpha hypothesis, you can use mathematical techniques to calculate beta. These techniques require additional information: an alpha value, a sample size and an effect size. The alpha value comes from your alpha hypothesis; it is the probability of type I error. The sample size is the number of data points in your data set. The effect size is usually estimated from past data.

List the values that are needed in the beta calculation. These values include alpha, the effect size and the sample size. If you do not have past data that states a clear effect size, use the value 0.3 to be conservative. Essentially, the effect size is the strength of the relationship in the data; thus 0.3 is usually taken as it is a “moderate” effect size.

Find the Z-score for the value 1 - alpha/2. This Z-score will be used in the beta calculation. After calculating the numerical value for 1 - alpha/2, look up the Z-score corresponding to that value. This is the Z-score needed to calculate beta.

Calculate the Z-score for the value 1 - beta. Divide the effect size by 2 and take the square root. Multiply this result by the effect size. Subtract the Z-score found in the last step from this value to arrive at the Z-score for the value 1 – beta.

Convert the Z-score to 1 - beta as a number. “Reverse” look up the Z-score for 1 - beta by first looking up the Z-score in the Z-table. Trace this Z-score back to the column (or row) to find a number. This number is equal to 1 - beta.

Subtract the number just found from 1. This result is beta.

• Virtually every introduction to statistics textbook has a Z-table in the appendix. If you do not have a Z-table on hand, consult a statistics book from your library.

Related Articles

How to calculate a two-tailed test, how to determine the sample size in a quantitative..., how to calculate cv values, what are the different types of correlations, how to calculate confidence levels, how to make a relative frequency table, how to know if something is significant using spss, how to calculate levered beta, how to calculate standard errors, how to calculate a t-statistic, the effects of a small sample size limitation, how to calculate statistical sample sizes, how to interpret a student's t-test results, how to calculate the percentage of another number, how to report a sample size, how to calculate bias, how to calculate the root mse in anova, how to calculate a confidence interval, how to write a hypothesis for correlation.

• “Essentials of Biostatistics”; Lisa Sullivan; 2008
• “Statistical Misconceptions”; Schuyler Huck; 2009

About the Author

Having obtained a Master of Science in psychology in East Asia, Damon Verial has been applying his knowledge to related topics since 2010. Having written professionally since 2001, he has been featured in financial publications such as SafeHaven and the McMillian Portfolio. He also runs a financial newsletter at Stock Barometer.

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Hypothesis testing.

Key Topics:

• Basic approach
• Null and alternative hypothesis
• Decision making and the p -value
• Z-test & Nonparametric alternative

Basic approach to hypothesis testing

• State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
• Specify the null and alternative hypotheses in terms of the parameters of the model.
• Invent a test statistic that will tend to be different under the null and alternative hypotheses.
• Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
• Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
• Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.
• \(\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}\)
• general form is: (estimate - value we are testing)/(st.dev of the estimate)
• z-statistic follows N(0,1) distribution
• 2 × the area above |z|, area above z,or area below z, or
• compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
• Choose the acceptable level of Alpha = 0.05, we conclude …. ?

Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

• either our initial assumption is correct and we experienced an unusual event or,
• our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

• H 0 : Defendant is not guilty.
• H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

• It can one-sided and two-sided.
• In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

• Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

• If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
• If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

• α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

• β is the probability of making a Type II error

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

• the effect of the sample size,
• the correctness of the underlying assumptions about the population,
• statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

One sample t -test

• Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .
• t-statistic: \(\frac{\bar{X}-\mu_0}{s / \sqrt{n}}\) where s is a sample st.dev.
• t-statistic follows t -distribution with df = n - 1
• Alpha = 0.05, we conclude ….

Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

In the brain, bursts of beta rhythms implement cognitive control

Bursts of brain rhythms with “beta” frequencies control where and when neurons in the cortex process sensory information and plan responses. Studying these bursts would improve understanding of cognition and clinical disorders, researchers argue in a new review.

The brain processes information on many scales. Individual cells electrochemically transmit signals in circuits but at the large scale required to produce cognition, millions of cells act in concert, driven by rhythmic signals at varying frequencies. Studying one frequency range in particular, beta rhythms between about 14-30 Hz, holds the key to understanding how the brain controls cognitive processes—or loses control in some disorders—a team of neuroscientists argues in a new review article.

Drawing on experimental data, mathematical modeling and theory, the scientists make the case that bursts of beta rhythms control cognition in the brain by regulating where and when higher gamma frequency waves can coordinate neurons to incorporate new information from the senses or formulate plans of action. Beta bursts, they argue, quickly establish flexible but controlled patterns of neural activity for implementing intentional thought.

“Cognition depends on organizing goal-directed thought, so if you want to understand cognition, you have to understand that organization,” said co-author Earl K. Miller , Picower Professor in The Picower Institute for Learning and Memory and the Department of Brain and Cognitive Sciences at MIT. “Beta is the range of frequencies that can control neurons at the right spatial scale to produce organized thought.”

Miller and colleagues Mikael Lundqvist, Jonatan Nordmark and Johan Liljefors at the Karolinska Institutet and Pawel Herman at the KTH Royal Institute of Technology in Sweden, write that studying bursts of beta rhythms to understand how they emerge and what they represent would not only help explain cognition, but also aid in diagnosing and treating cognitive disorders.

“Given the relevance of beta oscillations in cognition, we foresee a major change in the practice for biomarker identification, especially given the prominence of beta bursting in inhibitory control processes … and their importance in ADHD, schizophrenia and Alzheimer’s disease,” they write in the journal Trends in Cognitive Sciences .

Experimental studies covering several species including humans, a variety of brain regions, and numerous cognitive tasks have revealed key characteristics of beta waves in the cortex, the authors write: Beta rhythms occur in quick but powerful bursts; they inhibit the power of higher frequency gamma rhythms; and though they originate in deeper brain regions, they travel within specific locations of cortex. Considering these properties together, the authors write that they are all consistent with precise and flexible regulation, in space and time, of the gamma rhythm activity that experiments show carry signals of sensory information and motor plans.

“Beta bursts thus offer new opportunities for studying how sensory inputs are selectively processed, reshaped by inhibitory cognitive operations and ultimately result in motor actions,” the authors write.

For one example, Miller and colleagues have shown in animals that in the prefrontal cortex in working memory tasks, beta bursts direct when gamma activity can store new sensory information, read out the information when it needs to be used, and then discard it when it’s no longer relevant. For another example, other researchers have shown that beta rises when human volunteers are asked to suppress a previously learned association between word pairs, or to forget a cue because it will no longer be used in a task.

In a paper last year, Lundqvist, Herman, Miller and others cited several lines of experimental evidence to hypothesize that beta bursts implement cognitive control spatially in the brain , essentially constraining patches of the cortex to represent the general rules of a task even as individual neurons within those patches represent the specific contents of information. For example, if the working memory task is to remember a pad lock combination, beta rhythms will implement patches of cortex for the general steps “turn left,” “turn right,” “turn left again,” allowing gamma to enable neurons within each patch to store and later recall the specific numbers of the combination. The two-fold value of such an organizing principle, they noted, is that the brain can rapidly apply task rules to many neurons at a time and do so without having to re-establish the overall structure of the task if the individual numbers change (i.e. you set a new combination).

Another important phenomenon of beta bursts, the authors write, is that they propagate across long distances in the brain, spanning multiple regions. Studying the direction of their spatial travels, as well as their timing, could shed further light on how cognitive control is implemented.

New ideas beget new questions

Beta rhythm bursts can differ not only in their frequency, but also their duration, amplitude, origin and other characteristics. This variety speaks to their versatility, the authors write, but also obliges neuroscientists to study and understand these many different forms of the phenomenon and what they represent to harness more information from these neural signals.

“It quickly becomes very complicated, but I think the most important aspect of beta bursts is the very simple and basic premise that they shed light on the transient nature of oscillations and neural processes associated with cognition,” Lundqvist said.“This changes our models of cognition and will impact everything we do. For a long time we implicitly or explicitly assumed oscillations are ongoing which has colored experiments and analyses. Now we see a first wave of studies based on this new thinking, with new hypothesis and ways to analyze data, and it should only pick up in years to come.” 

The authors acknowledge another major issue that must be resolved by further research—How do beta bursts emerge in the first place to perform their apparent role in cognitive control?

“It is unknown how beta bursts arise as a mediator of an executive command that cascades to other regions of the brain,” the authors write.

The authors don’t claim to have all the answers. Instead, they write, because beta rhythms appear to have an integral role in controlling cognition, the as yet unanswered questions are worth asking.

“We propose that beta bursts provide both experimental and computational studies with a window through which to explore the real-time organization and execution of cognitive functions,” they conclude. “To fully leverage this potential there is a need to address the outstanding questions with new experimental paradigms, analytical methods and modeling approaches.”

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