Confusing Statistical Terms #2: Alpha and Beta

by Karen Grace-Martin

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 in the discussion section I was leading when one of the poor undergrads was hopelessly lost.  We were talking about the simple regression coefficient (beta) and the intercept (which the text we were using chose to call alpha, instead of the more familiar beta-naught).

It was only after repeated probing that I realized she was logically trying to fit it 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 error, 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.   Here they are:

Hypothesis testing

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

Alpha is the probability of Type I error in any hypothesis test–incorrectly claiming statistical significance.

Beta is the probability of Type II error in any hypothesis test–incorrectly concluding no statistical significance.  (1 – Beta is power).

Regression coefficients

In most textbooks and software packages, the population regression coefficients are denoted by beta.  Like all population parameters, they are theoretical–we don’t know what they are.  The regression coefficients we estimate from our sample are statistical 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 β0 (beta-naught–and yes, that’s the closest I can get to a subscript)  and every other regression coefficient as β1, β2, β3, etc.  But as I already mentioned, some statistics texts will refer to the intercept as alpha, to distinguish it from the other coefficients.

Standardized Regression Coefficients

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.

tn_mdWant to get up to speed on the meaning and logic of power, sample size, and how to calculate estimates? Check out our new On Demand online workshop called Calculating Power and Sample size.


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{ 11 comments… read them below or add one }

Andy 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.

Reply

Karen 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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Student 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!?!?!

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Karen 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 < .01 (usually one * for < .05 and two ** for < .01). But they're saying "alpha >“, “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?

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Carrie 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!

Reply

Karen March 25, 2011 at 12:54 pm

Thanks, Carrie! Glad it was helpful.

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Charlotte September 29, 2011 at 5:16 am

This is so helpful. Thx!!

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Anna July 30, 2012 at 1:46 am

Hi,

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?

Reply

Karen August 3, 2012 at 2:57 pm

Hi Anna,

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: http://www.theanalysisfactor.com/interpreting-the-intercept-in-a-regression-model/

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Ayesha July 5, 2013 at 10:51 am

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

Reply

Karen 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.

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