If possible, some mathematical formulations will help clarify the concept. This is a subtle question. It takes a thoughtful person not to understand those quotations! Although they are suggestive, it turns out that none of them is exactly or generally correct. I haven't the time and there isn't the space here to give a full exposition, but I would like to share one approach and an insight that it suggests.
Where does the concept of degrees of freedom DF arise? The contexts in which it's found in elementary treatments are:. The Student t-test and its variants such as the Welch or Satterthwaite solutions to the Behrens-Fisher problem where two populations have different variances. The Chi-squared distribution defined as a sum of squares of independent standard Normals , which is implicated in the sampling distribution of the variance.
The Chi-squared test , comprising its uses in a testing for independence in contingency tables and b testing for goodness of fit of distributional estimates.
This is of especial interest because it is the first hint that DF is not any of the things claimed of it. We can dispose right away of some of the claims in the question.
Because "final calculation of a statistic" is not well-defined it apparently depends on what algorithm one uses for the calculation , it can be no more than a vague suggestion and is worth no further criticism. Similarly, neither "number of independent scores that go into the estimate" nor "the number of parameters used as intermediate steps" are well-defined.
One is independence of random variables; the other is functional independence. The three side lengths can be considered independent random variables, but all five variables are dependent RVs. The five are also functionally dependent because the codomain not the "domain"! Having been alerted by these potential ambiguities, let's hold up the Chi-squared goodness of fit test for examination , because a it's simple, b it's one of the common situations where people really do need to know about DF to get the p-value right and c it's often used incorrectly.
Here's a brief synopsis of the least controversial application of this test:. It may be problematic when the bins are determined by the data, even though this is often done. Using these bins, the data are reduced to the set of counts within each bin. Equal-probability binning assures the chi-squared distribution really is a good approximation to the true distribution of the chi-squared statistic about to be described. You have a lot of data--enough to assure that almost all bins ought to have counts of 5 or greater.
Using the parameter estimates, you can compute the expected count in each bin. The Chi-squared statistic is the sum of the ratios. This, many authorities tell us, should have to a very close approximation a Chi-squared distribution.
But there's a whole family of such distributions. But there are functional relationships among them. That's one relationship. The problem with this reasoning which is the sort of calculation the quotations in the question are hinting at is that it's wrong except when some special additional conditions hold.
Moreover, those conditions have nothing to do with independence functional or statistical , with numbers of "components" of the data, with the numbers of parameters, nor with anything else referred to in the original question.
Let me show you with an example. To make it as clear as possible, I'm using a small number of bins, but that's not essential. To test goodness of fit, create four bins with cutpoints at the quartiles of a standard normal: Repeat as patience allows; I had time to do 10, repetitions. Here's the histogram:. Neither fits the data. However, the problem persists even with very large datasets and larger numbers of bins: it is not merely a failure to reach an asymptotic approximation.
You must use the Maximum Likelihood estimate of the parameters. This requirement can, in practice, be slightly violated. You must base that estimate on the counts, not on the actual data!
This is crucial. The red histogram depicts the chi-squared statistics for 10, separate iterations, following these requirements. The point of this comparison--which I hope you have seen coming--is that the correct DF to use for computing the p-values depends on many things other than dimensions of manifolds, counts of functional relationships, or the geometry of Normal variates.
There is a subtle, delicate interaction between certain functional dependencies, as found in mathematical relationships among quantities, and distributions of the data, their statistics, and the estimators formed from them.
Accordingly, it cannot be the case that DF is adequately explainable in terms of the geometry of multivariate normal distributions, or in terms of functional independence, or as counts of parameters, or anything else of this nature. We are led to see, then, that "degrees of freedom" is merely a heuristic that suggests what the sampling distribution of a t, Chi-squared, or F statistic ought to be, but it is not dispositive. Belief that it is dispositive leads to egregious errors. For instance, the top hit on Google when searching "chi squared goodness of fit" is a Web page from an Ivy League university that gets most of this completely wrong!
In particular, a simulation based on its instructions shows that the chi-squared value it recommends as having 7 DF actually has 9 DF. With this more nuanced understanding, it's worthwhile to re-read the Wikipedia article in question: in its details it gets things right, pointing out where the DF heuristic tends to work and where it is either an approximation or does not apply at all. I am grateful for the opportunity afforded by this question to lead me back to this wonderful text, which is full of such useful analyses.
Or simply: the number of elements in a numerical array that you're allowed to change so that the value of the statistic remains unchanged. The sketch of proof of these facts is given below. The two results are central for the further development of the statistical theory based on the normal distribution. I must admit that I don't find any of the paragraphs cited from the Wikipedia article particularly enlightening, but they are not really wrong or contradictory either.
Beyond the theory of linear normal models the use of the concept of degrees of freedom can be confusing. When we consider statistical analysis of categorical data there can be some confusion about whether the "independent pieces" should be counted before or after a tabulation.
Furthermore, for constraints, even for normal models, that are not subspace constraints, it is not obvious how to extend the concept of degrees of freedom. Various suggestions exist typically under the name of effective degrees of freedom.
Before any other usages and meanings of degrees of freedom is considered I will strongly recommend to become confident with it in the context of linear normal models.
A reference dealing with this model class is A First Course in Linear Model Theory , and there are additional references in the preface of the book to other classical books on linear models. This follows from general linear transformation results of the normal distribution. It's really no different from the way the term "degrees of freedom" works in any other field.
For example, suppose you have four variables: the length, the width, the area, and the perimeter of a rectangle.
Do you really know four things? No, because there are only two degrees of freedom. If you know the length and the width, you can derive the area and the perimeter. If you know the length and the area, you can derive the width and the perimeter. If you know the area and the perimeter you can derive the length and the width up to rotation.
If you have all four, you can either say that the system is consistent all of the variables agree with each other , or inconsistent no rectangle could actually satisfy all of the conditions.
A square is a rectangle with a degree of freedom removed; if you know any side of a square or its perimeter or its area, you can derive all of the others because there's only one degree of freedom. In statistics, things get more fuzzy, but the idea is still the same.
If all of the data that you're using as the input for a function are independent variables, then you have as many degrees of freedom as you have inputs.
It is, however, valid when estimating parameters using one sample. In the above example of satisfying the average, the sample size was equal to 3. Therefore, df for a sample size of three numbers would be:. T-tests go into calculating the average in hypothesis tests Hypothesis Tests Hypothesis Testing is the statistical tool that helps measure the probability of the correctness of the hypothesis result derived after performing the hypothesis on the sample data.
It confirms whether the primary hypothesis results derived were correct. If two samples collected are with different sizes, i. Let us assume samples gathered for the T-tests T-tests A T-test is a method to identify whether the means of two groups differ from one another significantly.
It is an inferential statistics approach that facilitates the hypothesis testing. Putting the values in the formula derived above for degrees of freedom for T test will give:. The chi-square test Chi-square Test In Excel, the Chi-Square test is the most commonly used non-parametric test for comparing two or more variables for randomly selected data.
It is a test that is used to determine the relationship between two or more variables. More importantly, the chi-square table uses df to determine the number of categorical variable data cells to calculate the values of other cells. It compares the row data with the column data to establish a relationship between two variables. In other words, each cell represents an observation or frequency for these variable inputs.
It also helps reject a hypothesis based on the number of variables and data samples available. For example, a medical center conducts a study to establish a relationship between gender and body fat percentage. It is where the chi-square test can help determine how two sets of categorical data are related.
The null hypothesis Null Hypothesis Null hypothesis presumes that the sampled data and the population data have no difference or in simple words, it presumes that the claim made by the person on the data or population is the absolute truth and is always right. So, even if a sample is taken from the population, the result received from the study of the sample will come the same as the assumption. On the other hand, the alternative approach would indicate the existence of a connection between two variables.
Let us move ahead with the abovementioned example to find out the df. The set of observations obtained by the medical center is as follows:. It doesn't matter what values you use for the row and column marginal totals. Once those values are set, there's only one cell value that can vary here, shown with the question mark—but it could be any one of the four cells.
Once you enter a number for one cell, the numbers for all the other cells are predetermined by the row and column totals. They're not free to vary. So the chi-square test for independence has only 1 degree of freedom for a 2 x 2 table. Similarly, a 3 x 2 table has 2 degrees of freedom, because only two of the cells can vary for a given set of marginal totals.
For a table with r rows and c columns, the number of cells that can vary is r -1 c The degrees of freedom then define the chi-square distribution used to evaluate independence for the test. The chi-square distribution is positively skewed. As the degrees of freedom increases, it approaches the normal curve. Degrees of freedom is more involved in the context of regression. Rather than risk losing the one remaining reader still reading this post hi, Mom!
Recall that degrees of freedom generally equals the number of observations or pieces of information minus the number of parameters estimated. When you perform regression, a parameter is estimated for every term in the model, and and each one consumes a degree of freedom.
Therefore, including excessive terms in a multiple regression model reduces the degrees of freedom available to estimate the parameters' variability. In fact, if the amount of data isn't sufficient for the number of terms in your model, there may not even be enough degrees of freedom DF for the error term and no p-value or F-values can be calculated at all.
You'll get output something like this:. If this happens , you either need to collect more data to increase the degrees of freedom or drop terms from your model to reduce the number of degrees of freedom required. So degrees of freedom does have real, tangible effects on your data analysis, despite existing in the netherworld of the domain of a random vector. This post provides a basic, informal introduction to degrees of freedom in statistics.
If you want to further your conceptual understanding of degrees of freedom, check out this classic paper in the Journal of Educational Psychology by Dr. Helen Walker, an associate professor of education at Columbia who was the first female president of the American Statistical Association.
Another good general reference is by Pandy, S. Minitab Blog. What Are Degrees of Freedom in Statistics? Minitab Blog Editor 08 April, The Freedom to Vary First, forget about statistics. Degrees of Freedom: 1-Sample t test Now imagine you're not into hats.
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