What Are Degrees Of Freedom?

are the same (that is,fe-1 2 - 1 \,N-k 102- 2 100) as the degrees offireedomassociated with"equal" variance test discussed earlier,and thereforeSPSS does not report it separately.If the assumption of equal variance is violatedand the two groups have different variances as isthe case in this example, where the folded F test orLevene's F weighted statistic is significant, indicating that the two groups have significantly differentvariances, the value for degrees of freedom (100) isno longer accurate.Therefore, we need to estimatethe correct degrees of freedom (SAS Institute, 1985;also see Satterthwaite, 1946, for the computationsinvolved in this estimation).We can estimate the degrees of freedom according to Satterthwaite's (1946) method by using thefollowing formula:(//"Satterthwaite (H-])(»,-!)(« - 1 ) 1 where n - sample size of group \,n sample sizeof group 2, and S and S are the standard deviationsof groups 1 and 2,respectiveIy.By inserting subgroupdata from Table 1, we arrive at the more accuratedegrees of freedom as follows:(64-1) (38-1)25.65 X 38(64-1) 1 - / 25.65 X 3825.65- X 38 (38-1) / 25.65 X 38V 18.07 X 64,U 18.07 x64,233163 1 -25000.9625000.96h 20897.59; 3725000.9625000.96& 20897.59,2331.7 2 5 0(i(X9645898.55J'''[45898.55233163[1 - .5447] value for degrees of freedom, 96.97 (SAS rounds itto 97), is accurate, and our earlier value, 100, is not.Fortunately, it is no longer necessary to hand calculate this as major statistical packages such as SASand SPSS provide the correct value for degrees offreedom when the assumption of equal variance isviolated and equal variances are not assumed. Thisis the fourth value for degrees of freedom in ourexample, which appears in Table 1 as 97 in SASand 96.967 in SPSS. Again, this value is the correctnumber to report, as the assumption of equal variances is violated in our example.Comparing the Means of g Groups withOne Parameter (Analysis of Variance)What if we have more than two groups to compare? Let us assume that we have «, . . . «groups of observations or countries grouped bypohtical freedom (FP EDOMX) and that weare interested in differences in their literacy rates(LITERACY, the dependent variable ).We can testthe variability of means by using the analysis ofvariance (ANOVA).Thc ANOVA procedure produces three different types ot degrees of freedom,calculated as follows: The first type of degrees of freedom is calledthe between-groups degrees of freedom or modeldegrees of freedom and can be determined byusing the number of group means we wantto compare. The ANOVA procedure teststhe assumption that the g groups have equalmeans and that the population mean is notstatistically different from the individual groupmeans. This assumption reflects the null hypothesis, which is that there is no statisticallysignificant difference between literacy ratesin g groups of countries (ji fX ).Thealternative hypothesis is that the g samplemeans are significantly different from oneanother. There are g - 1 model degrees offreedom for testing the null hypothesis and forassessing variability among the means.Thisvalue of model degrees of freedom is used inthe numerator for calculating the F ratio inANOVA. The second type of degrees offreedom,called233163 X .207 37 X .29672331 96.9724.0379Because the assumption of equality of variances isviolated, in the previous analysis the Satterthwaite's122the within-groups degrees of freedom or error de-grees offreedom, is derived from subtracting themodel degrees offreedom from the correctedtotal degrees of freedom. The within-groupsSocial Work Research VOLUME 32, NUMBER 2 JUNE 2008

degrees of freedom equal the total numberof observations minus the number of groupsto be compared, n . . n -g. This valuealso accounts for the denominator degrees offreedom for calculating the F statistic in anANOVA. Calculating the third type of degrees offreedom is straightforward. We know thatthe sum of deviation from the mean or 2(Y- F) O.We also know that the total sum ofsquares or 2(y - Y) is nothing but the sumof N- deviations from the mean. Therefore,to estimate the total sum of squares 2(y ?)-, we need only the sum of N - 1 deviations from the mean.Therefore.with the totalsample size we can obtain the total degreesof freedom, or corrected total degrees offreedom, by using the formula N - 1.hiTable 2, we show the SAS and SPSS output withthese three different values of degrees of freedomusing the ANOVA procedure.The dependent variable.literacy rate.is continuous,and the independentvariable, political freedom or FREEDOMX, isnominal. Countries are classified into three groupson the basis of the amount of political freedom eachcountry enjoys: Countries that enjoy high politicalfreedom are coded as 1 (« 32), countries that enjoymoderate political freedom are coded as 2 (n 34),and countries that enjoy no pohticai freedom arecoded as 3 (tt 36). The mean literacy rates (dependent variable) of these groups of countries areexamined.The null hypothesis tests the assumptionchat there is no significant difference in the literacyr.ites of these countries according to their level ofpolitical freedom.The first of the three degrees of freedom, thebetween-groups degrees of freedom, equals g - \.Because there are three groups of countries in thisanalysis, we have 3 - 1 2 degrees of freedom.Thisaccounts for the numerator degrees ot freedom inestimating the Fsta tis tic. Second, the wi thin-groupsdegrees of freedom, which accounts for the denominator degrees of freedom for calculating the Fstatistic in ANOVA, equals ri . . n -g. Thesedegrees of freedom are calculated as 32 34 36- 3 99. Finally, the third degrees of freedom, thetotal degrees of freedom, are calculated as N - 1(102-1 101).When reporting Fvalues and theirrespective degrees of freedom, researchers shouldreport them as follows: The independent and thePANDEY AND BRtGHT / WhotArc Dtgrcfs ofFrefdom?dependent variables are significantly related [F(2,99) 16.64,p .0001].Degrees of Freedom in MultipleRegression AnalysisWe skip to multiple regression because degrees offreedom are the same in ANOVA and in simpleregression. In multiple regression analysis, there ismore than one independent variable and one dependent variable. Here, a parameter stands for therelationship between a dependent variable (Y) andeach independent variable (X). One must understand four different types of degrees of freedom inmultiple regression. The first type is the model (regremoti) degreesof freedom. Model degrees of freedom are associated with the number of independentvariables in the model and can be understoodas follows:A null model or a model without independent variables will have zero parameters tobe estimated. Therefore, predicted V* is equalto the mean of Vand the degrees of freedomequal 0.A mode! with one independent variable hasone predictor or one piece of useful information (fe 1) for estimation of variability in Y.This model must also estimate the point wherethe regression line originates or an intercept.Hence, in a model with one predictor, thereare (fe t) parameters—k regression coefficients plus an intercept—to be estimated,with k signifying the number of predictors.Therefore,there are \{k ])- l],orfe degreesof freedom for testing this regression model.Accordingly, a multiple regression modelwith more than one independent variable hassome more useful information in estimatingthe variability in the dependent variable, andthe model degrees of freedom increase as thenumber of independent variables increase.Thenull hypothesis is that all of the predictonhave the same regression coefficient of zero,thus there is otily one common coefficientto be estimated (Dallal,2(X)3).The alternativehypothesis is that the regression coefficientsare not zero and that each variable explains adifferent amount of variance in the dependentvariable. Thus, the researcher must estimatefe coefficients plus the intercept. Therefore,123

rs *fNlA00frtlA OlA(TVOrjr 784.3! 31.000728.84912126.74!Subset (a .05)OIIIIIIf (N—1mbeiticalMlvi1 FREEDOMXBetween GroupsWithin GroupsTotalSum ofSquaresCVII i"0 t:a 'Sy ISi aI 3o3J2il16.638.000

there are (fe 1) - 1 or it degrees offreedomtor testing the null hypothesis (Dallal, 2003).In other words, the model degrees offreedomequal the number of useful pieces ofinformacioii available for estimation of variability inthe dependent variable.1. as explained above. F values and the respectivedegrees of freedom from the current regressionoutput should be reported as follows: Tbe regression model is statistically significant with F(6, 92) 44.86,p .0001.The second type is tbe residual, or error, degreesoffreedom. Residual degrees of freedom in multiple regression involve information of bothsample size and predictor variables. In addition,we also need to account for the intercept. Forexample, if our sample size equals N, we needto estimate k l parameters, or one regressioncocfiU-ient for each of the predictor variables{k) plus one for the intercept. The residualdegrees of freedom are calculated N - {k l).This is the same as the formula for theerror, or within-groups, degrees offreedomin the ANOVA. It is important to note thatincreasing the number of predictor variableshas implications for the residual degrees offreedom. Each additional parameter to beestimated costs one residual degree offreedom{Dallal,2()03).The remaining residual degreesoffreedom are used to estimate variability inthe dependent variable.Degrees of Freedom in aNonparametric TestTbe third type of degrees offreedom is thetotal, o r corrected total, degrees offreedom. As in ANOVA, this is calculated N- \.Finally, the fourth rype of degrees offreedomthat SAS (and not SPSS) reports under theparameter estimate in multiple regression isworth mentioning. Here, the null hypothesisis that there is no relationship between eachindependent variable and the dependent variable. The degree of freedom is always t foreach relationship and therefore, some statisticalsofrware.sucb as SPSS,do not bother to reportit.In the example of multiple regression analysis (seeTable 3), there are four different values of degreesof freedom. The first is the regression degrees offreedom. This is estimated as (fe 1) - 1 or (6 1) - 1 6 , where k is the number of independentvariables in the model. Second, the residual degreesoffreedom are estimated as N- {k 1). Its valuehere is 99 - (6 1) 92.ThinJ, the total degreesoffreedom are calculated N - 1 (or 99 -1 98).Finally, the degrees of freedom shown under parameter estimates for each parameter always equalI'ANDEV AND BRIGHT / What Are Degrees ofIivedom?Pearson's chi square, or simply tbe chi-square statistic, is an example ofa nonparametric test that iswidely used to examine the association between tvi'onominal level variables. According to Weiss (1968)"the number of degrees offreedom to be associatedwith a chi-square statistic is equal to the number ofindependent components that entered into its calculation" (p. 262). He further explained that each cell ina chi-square statistic represents a single componentand that an independent component is one whereneither observed nor expected values are determinedby the frequencies in other cells. In other vfords, ina contingency table, one row and one column arefated and tbe remaining cells are independent and arefree to vary. Therefore, the chi-square distributionhas ( r - 1) X ( c - 1) degrees of freedom, where r isthe number of rows and f is tbe number of columnsin the analysis (Cohen, 1988; Walker, 1940; Weiss,1968). We subtract one from both the number ofrows and columns simply because by knowing thevalues in other cells we can tell the values in the lastcells for both rows and columns; therefore, these lastcells are not independent.\As an example, we ran a chi-square test to exatninewhether gross national product (GNP) per capitaof a country (GNPSPLIT) is related to its levelof political freedom (FREEDOMX). Countries(GNPSPLIT) are divided into two categories—richcountries or countries with higji GNP per capita(coded as 1) and poor countries or countries withlow GNP per capita (coded as O),and political freedom (FREEDOMX) has three levels—free (codedas 1), partly free (coded as 2), not free (coded as 3)(see Table 4). In this analysis, the degrees offreedom arc (2 - 1) X (3 - 1) 2. In other words, byknowing the number of rich countries, we wouldautomatically know the number of poor countries.But by knowing the number of countries that arefree, we would not know the number of countriesthat are partly free and not free. Here, we need toknow two of the three components—for instance,the number of countries that are free and partlyfree—so that we will know the number of countries125

coo44.86100oofSII PlSCHC0w00c.ZBdel—-ii-ON « (N—"A »o.20.08o—ooV00INI/Nt— } -Is.PJJPSICINS so1/5111—IT p(SCN-V00o o oJ4 .S S E E33aZ Z2t §B -02 -.r(N *jCNi IN00OOw q.g r (*) \oq.'I - OON—so—mOoVV92o.39o.2-aSc2c5q— r 1soCOop —1tVari-§00GNP21VaricSum v cD00rNpan Uu ""1q(ConUJ-oeffiQ00CN00ofSqsofNONqCNQcc00(N"T(N p.ooq ? op3.21 Standa:1 Coeffi. enrs262.17711761.62016.1919cs(N" oqsoo1/*sov ,729Estimate2Adjusted / '00

Pearson X"Likelihood RatioLinear-by-Linear AssociationValid Cases (N)Total1Count% of TotalCount% of TotalCount% of al192Asymptotic, PLIT-aFREEDOMX64.6 GNPSPLIT FREEDOMXMissing100.0nfN(N —oooooo3u

that are notfree.Therefore,in this analysis there aretwo independent components that are free to vary,and thus the degrees of freedom are 2.Readers may note that there are three values underdegrees of freedom in Table 4. The first two valuesare calculated the same vi'ay as discussed earlier andhave the same values and are reported most widely.These are the values associated with the Pearsonchi-square and likelihood ratio chi-square tests.Thefmal test is rarely used. We explain this briefly.Thedegree of freedom for the Mantel-Haenszel chi- square statistic is calculated to test the hypothesisthat the relationship between two variables (rowand column variables) is linear; it is calculated as(N " 1) X f , where r is the Pearson product-moment correlation between the row variable and thecolumn variable (SAS Institute. 1990).This degreeof freedom is always 1 and is useful only when bothrow and column variables are ordinal.CONCLUSIONYu (1997) noted that "degree of freedom is anintimate stranger to statistics students" (p. l).Thisresearch note has attempted to decrease the strangeness of this relationship with an introduction to thelogic of the use of degrees of freedom to correctlyinterpret statistical results. More advanced researchen, however, will note that the information providedin this article is limited and fairly elementary. Asdegrees of freedom vary by statistical test (Salkind,2004), space prohibits a more comprehensive demonstration. Anyone with a desire to learn moreabout degrees offreedomin statistical calculations isencouraged to consult more detailed resources,suchas Good (1973),Walker (1940),andYu (1997).Finally, for illustrative purposes we used WorldData that reports information at country level. Inour analysis, we have treated each country as anindependent unit of analysis. Also, in the analysis,each country is given the same weight irrespective of its population size or area. We have ignoredlimitations that are inherent in the use of such data.We warn readers to ignore the statistical findings ofour analysis and take away only the discussion thatpertains to degrees of feedom.Healey.J. F. (1990). Statistics: A tool for social research (2nded,), Belmont, CA: Wadsworth,HyperStat Onlitie. (n,d,). Degrees of freedom. Retrieved May30, 2006, from ,J., & Becker, M. A. (1990). Statistiafor the behavioralsciences (2nd ed.), Belmont, CA:Wadsworth,Norman. G. R,, & Streiner, D. L, (20113), PDQ statistics (3rded,). Hamilton, Ontario, Canada: BC Decker,Pagano. R, R, (2004), Understanding statistics in the behavioralsciences (7th ed,). Belmont, CA: Wadsworth,Rosenthal, j.A, (2001), Statistics and data interpretaliou for thehelping professions. Belmont. CA: Wadsworth.Runyon. R. P,,& Haber,A. (1991). Fundamcntak of behavioral statistics (7th ed.). New York: McGraw-Hill,Salkind, N.J, (21)04), Statistics for people who (think they)hate statistics (2nd ed.).Thousand Oaks, CA: SagePublications,SAS Institute. (1985). SAS user's guide: Statistics, version 5.1.Cary, NC: SAS Institute.SAS Institute, (1990), SAS procedure uide, version 6 (3nJed.). Cary, NC: SAS Institute, 'Satterthwaite, F. E. (1946), An approximate distribution ofestimates of variance components. Biometrics Bulletin,2, 110-114.Toothaker, L, E., &: MiUer, L, (1996), Introductory statisticsfor the behavioral sciences (2nd ed.). Pacific Grove, CA:Brooks/Cole,Trochim,W.M.K. (2005), Research methods:The conciseknowledge base. Cincinnati: Atomic Dog,Walker, H,W, (1940), Degrees of freedom. Hrrti/ ofEdiuational Psychology, 31, 2 5 3 - 2 6 9 .Warner. R. M. (2008). Applied sMti.(fi«,Thousand Oaks,CA: Sage Publications,Weinbach, R. W., & Griniiell, R. M.,Jr. (2004). Statistics forsocial workers (6th ed.). Boston: Pearson.Weiss, R. S. (1968). Statistics in social research:An introdudion.New York: John Wiley & Sons.Wonnacott,T, H., & Woiinacott, R.J, (1990). Introductorystatistics (5th ed,). New York: John Wiley & Sons.Yu, C. H, (1997), Illustrating degrees of freedom in terms ofsample size and dimensionality. Retrieved November1,2007, from nnlShanta Pandey, PhD, is associate profes. :or, and CharlotteLyn Bright, MSH is a doctoral student, George Warren BrownSchool of Social Work, Washington University, St. Louis. Theauthors are thankful to a student whose curious mind inspiredthem to work on this research note. Correspondence concerningthis article should be sent to Shanta Pcindey, Geotge WarrenBrown School of Social Work, Washington University, St. Louis,MO 63130; e-mail:parideYs@unistl.edu.Original manuscript received August 29, 2006Final revision received November 15, 2007Accepted January 16, 2008REFERENCESCoheti,J, (1988), Statistical power analysis for the behavioralsciences (2nd ed,), Hillsdale, NJ: Lawrence Eribauni.Dallal. G. E, (2003). Decrees of freedom. Retrieved May 30,2006. trotn http;//www.tufts,edu/ gdailal/dof,hemGood. LJ. (1973), What are degrees of freedom? AmerkimStatistician, 21, 227-228.128Social Work Research VOLUME 32, NUMBER 2JUNE 1008

of degrees of freedom and reports folded F statistics. The numerator degrees of freedom are calculated as n — 1, that is 64 — 1 63. The denominator degrees of freedom are calculated as n - 1 or 38 - 1 37.These degrees of freedom ar