ANOVA+Table+Formats

ANOVA Table Formats (Analysis of Variance | Regression} (GWU EMSE-271)
Index | Topics (Logical Lectures) | Lectures | Problems | Readings | Nomenclature | Concepts


 * Tips and Notes:**
 * 1) Sum of Squares and Degrees of Freedom sum to Total.
 * 2) Tried to corrected VanDorp SSsubsripts errors and move to Lattin convention for some subsripts (2-Way ANOVA "error" term.)
 * 3) Inputs for Signifance (F-Test) are degrees of free to left, degrees of freedom of "factor" divided by degrees of freedom of residuals.
 * 4) Tried to add parameters for p-value tests; worth someone checking.

**Regression Analysis** (ANOVA is a type of regressions analysis; regression analysis is not ANOVA {although variances are analyzed}. Table put here because it and the concepts to evaluate are so similar.) n = observations; p = number of variables; degrees of freedom of residual = n-p-1 becuase of the intercept; df of regession (difference) = p perhaps because should be p-1 (but counting intercept or p if p is not used to count intercept {(p+1}-1}; p for practical EMSE 271 practical purposes.
 * **Source** || Sum of Squares || Degrees of Freedom || Mean Square || **Fo** || p-value **(**F distribution) ||
 * Regression || SSregression || p (+ 1 {intercept}) -1 ??? || SSregression / df || MS regression / MS residuals || F(Fo, "p", n-p-1) or F (df(regression, df(residuals) ||
 * Residual || SSresiduals || n-p-1 || SSresidual / df ||  ||   ||
 * Total || SS(capT) || n-1 ||  ||   ||   ||

**One-Way ANOVA** [1] if P-value is low, at least ONE treatment differs. Comment probably belongs some where else, but put here temporarily. [Slide 305) Shows conclusions for interpreting contrasts, conclusions are based on the pre-defined Hypotheses (Slide 304) MSE = sigma^2 = SSE/ab(n-1)
 * **Source** || Sum of Squares || Degrees of Freedom || Mean Square || **Fo** || p-value (F distribution) ||
 * Between Treatments || SStreatments || p-1 || MStreatment || MStreatments/MSe || F(p-1,N-p) or F(df (Treatments), df (e)) ||
 * Error (within treatments) || SSe || N-p (np-p) || MSe ||  ||   ||
 * Total || SS(capT) || N-1 (np-1) ||  ||   ||   ||
 * ** Contrasts ** ||  ||   ||   ||   ||   ||
 * Contrast || SSc || 1 || MSc = SSc ||  || F (1,N-p) ||

p = number of treatements; n = observations (runs or tests) per treatment (if equal observations per treatment); N = n*p

**Two-Way ANOVA** Lattin, Table 11.5, page 404 and EMSE 271, Fall 2009, Slide 318. a is rows, b is columns, n is the number of trails for each "factor" (rows and columns, factor and method, etc.) Note: Van Dorp usess SSe and MSe, Lattin, I think, uses SSw and MSw, which I thought seemed better theoretically when I constructed this page. - pws
 * **Source** || Sum of Squares || Degrees of Freedom || Mean Square || **Fo** || p-value **(**F-distribution (df1,df2) ||
 * Factor A || SSa || a-1 || SSa / df(a) || MSa / MSw || F (a-1, ab(n-1)) or F(df (a), df (w)) ||
 * Factor B || SSb || b-1 || SSb / df(b) || MSb / MSw || F (b-1, ab(n-1)) or F (df (b), df (w)) ||
 * AB Interaction || SSab || (a-1)(b-1) || SSab / df(ab) || MSab / MSw || F ((a-1)(b-1), ab(n-1) or F( df (ab), df (w)) ||
 * Error || SSw || ab(n-1) || SSe / df (w)(error) ||  || Note: within-group sum of squares ||
 * Total || SS(capT) || abn-1 ||  ||   ||   ||

​a is number of scenarios (types of tests) for factor a; b is number of scenarios for factor b; n is number of observations per cell (if equal observations per cell)

**2(to the)K** **Factorial ANOVA** (General Format) a and b have different definitions than in 2-Way table. 4 is 2 to 2 (2 to the K) for the sample on slide 327 and n is the number of replications (often columns)
 * **Source** || Sum of Squares || Degrees of Freedom || Mean Square || **Fo** || p-value **(F-distribution)** ||
 * Factor A || ([a+ab-b-(1)]*2)/4n || 1 || SSa || MSa / MSe || F (1, 4(n-1)) or F (df (a), df (e)) ||
 * Factor B || ([b+ab-a-(1)]*2)/4n || 1 || SSb || MSb / MSe || F (1, 4(n-1)) or F (df (b), df (e)) ||
 * AB Interaction || ([ab+1-a-b]*2)/4n || 1 || SSab || MSab / MSe || F (1, 4(n-1) or F( df (ab), df (e)) ||
 * Error || SSt-SSa-SSb-SSab || 4(n-1) || SSe / df (e)(error) ||  ||   ||
 * Total || See Slide 327 || 4n-1 ||  ||   ||   ||


 * Sources:**
 * Analyzing Multivariate Data, by James Lattin, Douglas Carroll and Green, (c) 2003 ([|Amazon])
 * EMSE 271, Fall 2009