naginterfaces.library.anova.hier2

naginterfaces.library.anova.hier2(y, lsub, nobs)

hier2 performs an analysis of variance for a two-way hierarchical classification with subgroups of possibly unequal size, and also computes the treatment group and subgroup means. A fixed effects model is assumed.

For full information please refer to the NAG Library document for g04ag

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g04/g04agf.html

Parameters

yfloat, array-like, shape $\left(n\right)$

The elements of $y$ must contain the observations $y_{mij}$ in the following order:

$$y_{111},y_{211},\dots ,y_{n_{11}11},y_{112},y_{212},\dots ,y_{n_{12}12},\dots ,y_{11l_1},\dots \text{,}$$

$$y_{n_{1l_1}1l_1},\dots ,y_{1ij},\dots ,y_{n_{ij}ij},\dots ,y_{1k{l}_k},\dots ,y_{n_{k{l}_k}k{l}_k}\text{.}$$

In words, the ordering is by group, and within each group is by subgroup, the members of each subgroup being in consecutive locations in $y$.

lsubint, array-like, shape $\left(k\right)$

The number of subgroups within group $\text{i}$, $l_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

nobsint, array-like, shape $\left(l\right)$

The numbers of observations in each subgroup, $n_{ij}$, in the following order:

$$n_{11},n_{12},\dots ,n_{1l_1},n_{21},\dots ,n_{2l_2},\dots ,n_{k1},\dots ,n_{k{l}_k}$$

Returns

ngpint, ndarray, shape $\left(k\right)$

The total number of observations in group $\text{i}$, $n_{\text{i}.}$, for $\text{i}=1,2,\dots ,k$.

gbarfloat, ndarray, shape $\left(k\right)$

The mean for group $\text{i}$, ${\overline{y}}_{.\text{i}.}$, for $\text{i}=1,2,\dots ,k$.

sgbarfloat, ndarray, shape $\left(l\right)$

The subgroup means, ${\overline{y}}_{.ij}$, in the following order:

$${\overline{y}}_{.11},{\overline{y}}_{.12},\dots ,{\overline{y}}_{{.1l}_1},{\overline{y}}_{.21},{\overline{y}}_{.22},\dots ,{\overline{y}}_{{.2l}_2},\dots ,{\overline{y}}_{.k1},{\overline{y}}_{.k2},\dots ,{\overline{y}}_{{.kl}_k}\text{.}$$

gmfloat

The grand mean, ${\overline{y}}_{\dots }$.

ssfloat, ndarray, shape $\left(4\right)$

Contains the sums of squares for the analysis of variance, as follows;

$ss\left[0\right]=$ Between group sum of squares, ${ss}_g$,

$ss\left[1\right]=$ Between subgroup within groups sum of squares, ${ss}_{sg}$,

$ss\left[2\right]=$ Residual sum of squares, ${ss}_{res}$,

$ss\left[3\right]=$ Corrected total sum of squares, ${ss}_{tot}$.

idfint, ndarray, shape $\left(4\right)$

Contains the degrees of freedom attributable to each sum of squares in the analysis of variance, as follows:

$idf\left[0\right]=$ Degrees of freedom for between group sum of squares,

$idf\left[1\right]=$ Degrees of freedom for between subgroup within groups sum of squares,

$idf\left[2\right]=$ Degrees of freedom for residual sum of squares,

$idf\left[3\right]=$ Degrees of freedom for corrected total sum of squares.

ffloat, ndarray, shape $\left(2\right)$

Contains the mean square ratios, $F_1$ and $F_2$, for the between groups variation, and the between subgroups within groups variation, with respect to the residual, respectively.

fpfloat, ndarray, shape $\left(2\right)$

Contains the significances of the mean square ratios, $p_1$ and $p_2$ respectively.

Raises

NagValueError

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k>1$.

(errno $2$)

On entry, $i=⟨value⟩$ and $lsub[i-1]=⟨value⟩$.

Constraint: $lsub[i-1]>0$.

(errno $3$)

On entry, ${\sum }_{i}lsub\left[i\right]=⟨value⟩$ and $l=⟨value⟩$.

Constraint: ${\sum }_{i}lsub\left[i\right]=l$.

(errno $4$)

On entry, $i=⟨value⟩$ and $nobs[i-1]=⟨value⟩$.

Constraint: $nobs[i-1]>0$.

(errno $5$)

On entry, ${\sum }_{i}nobs[i-1]=⟨value⟩$ and $n=⟨value⟩$.

Constraint: ${\sum }_{i}nobs[i-1]=n$.

Warns

NagAlgorithmicWarning

(errno $6$)

The total corrected sum of squares is zero.

(errno $7$)

The residual sum of squares is zero.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

In a two-way hierarchical classification, there are $k$ ($\ge 2$) treatment groups, the $i$th of which is subdivided into $l_i$ treatment subgroups. The $j$th subgroup of group $i$ contains $n_{ij}$ observations, which may be denoted by

$$y_{1ij},y_{2ij},\dots ,y_{n_{ij}ij}\text{.}$$

The general observation is denoted by $y_{mij}$, being the $m$th observation in subgroup $j$ of group $i$, for $1\le i\le k$, $1\le j\le l_i$, $1\le m\le n_{ij}$.

The following quantities are computed

1. The subgroup means

   $${\overline{y}}_{.ij}=\frac{{\sum }_{m=1}^{n_{ij}}{y}_{mij}}{n_{ij}}$$

2. The group means

   $${\overline{y}}_{.i.}=\frac{{\sum }_{j=1}^{l_i}{\sum }_{m=1}^{n_{ij}}{y}_{mij}}{{\sum }_{j=1}^{l_i}{n}_{ij}}$$

3. The grand mean

   $${\overline{y}}_{\dots }=\frac{{\sum }_{i=1}^k{\sum }_{j=1}^{l_i}{\sum }_{m=1}^{n_{ij}}{y}_{mij}}{{\sum }_{i=1}^k{\sum }_{j=1}^{l_i}{n}_{ij}}$$

4. The number of observations in each group

   $$n_{i.}=\sum _{j=1}^{l_i}{n}_{ij}$$

5. Sums of squares

   Between groups	$={SS}_g={\sum }_{i=1}^k{n}_{i.}{({\overline{y}}_{.i.}-{\overline{y}}_{\dots })}^2$
   Between subgroups within groups	$={SS}_{sg}={\sum }_{i=1}^k{\sum }_{j=1}^{l_i}{n}_{ij}{(y_{.ij}-{\overline{y}}_{.i.})}^2$
   Residual (within subgroups)	$={SS}_{res}={\sum }_{i=1}^k{\sum }_{j=1}^{l_i}{\sum }_{m=1}^{n_{ij}}{(y_{mij}-{\overline{y}}_{.ij})}^2={SS}_{tot}-{SS}_g-{SS}_{sg}$
   Corrected total	$={SS}_{tot}={\sum }_{i=1}^k{\sum }_{j=1}^{l_i}{\sum }_{m=1}^{n_{ij}}{(y_{mij}-{\overline{y}}_{\dots })}^2$

6. Degrees of freedom of variance components

   Between groups:	$k-1$
   Subgroups within groups:	$l-k$
   Residual:	$n-l$
   Total:	$n-1$

   where

   $l={\sum }_{i=1}^k{l}_i$,

   $n={\sum }_{i=1}^k{n}_{i.}$

7. $F$ ratios. These are the ratios of the group and subgroup mean squares to the residual mean square.

   Groups	$F_1=\frac{\text{Between groups sum of squares}/(k-1)}{\text{Residual sum of squares}/(n-l)}=\frac{{SS}_g/(k-1)}{{SS}_{res}/(n-l)}$
   Subgroups	$F_2=\frac{\text{Between subgroups (within group) sum of squares}/(l-k)}{\text{Residual sum of squares}/(n-l)}=\frac{{SS}_{sg}/(l-k)}{{SS}_{res}/(n-l)}$

   If either $F$ ratio exceeds $9999.0$, the value $9999.0$ is assigned instead.

8. $f$ significances. The probability of obtaining a value from the appropriate $F$-distribution which exceeds the computed mean square ratio.

   Groups	$p_1=Prob(F_{(k-1),(n-l)}>F_1)$
   Subgroups	$p_2=Prob(F_{(l-k),(n-l)}>F_2)$

   where $F_{{\nu }_1,{\nu }_2}$ denotes the central $F$-distribution with degrees of freedom ${\nu }_1$ and ${\nu }_2$.

   If any $F_i=9999.0$, then $p_i$ is set to zero, $i=1,2$.

References

Kendall, M G and Stuart, A, 1976, The Advanced Theory of Statistics (Volume 3), (3rd Edition), Griffin

Moore, P G, Shirley, E A and Edwards, D E, 1972, Standard Statistical Calculations, Pitman