Maggie J. Xiong > PDL-Stats-0.6.3_3 > PDL::Stats::GLM

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# NAME

PDL::Stats::GLM -- general and generalized linear modeling methods such as ANOVA, linear regression, PCA, and logistic regression.

# DESCRIPTION

The terms FUNCTIONS and METHODS are arbitrarily used to refer to methods that are threadable and methods that are NOT threadable, respectively. FUNCTIONS except ols_t support bad value. PDL::Slatec strongly recommended for most METHODS, and it is required for logistic.

P-values, where appropriate, are provided if PDL::GSL::CDF is installed.

# SYNOPSIS

```    use PDL::LiteF;
use PDL::NiceSlice;
use PDL::Stats::GLM;

# do a multiple linear regression and plot the residuals

my \$y = pdl( 8, 7, 7, 0, 2, 5, 0 );

my \$x = pdl( [ 0, 1, 2, 3, 4, 5, 6 ],        # linear component
[ 0, 1, 4, 9, 16, 25, 36 ] );   # quadratic component

my %m  = \$y->ols( \$x, {plot=>1} );

print "\$_\t\$m{\$_}\n" for (sort keys %m);```

Replaces bad values with sample mean. Mean is set to 0 if all obs are bad. Can be done inplace.

```     perldl> p \$data
[
]

perldl> p \$data->fill_m
[
[      5     3.5       2     3.5]
[      7       3       7 5.66667]
] ```

Replaces bad values with random sample (with replacement) of good observations from the same variable. Can be done inplace.

```    perldl> p \$data
[
]

perldl> p \$data->fill_rand

[
[5 2 2 5]
[7 3 7 7]
]```

Replaces values with deviations from the mean. Can be done inplace.

Standardize ie replace values with z_scores based on sample standard deviation from the mean (replace with 0s if stdv==0). Can be done inplace.

Sum of squared errors between actual and predicted values.

Mean of squared errors between actual and predicted values, ie variance around predicted value.

Root mean squared error, ie stdv around predicted value.

Calculates predicted prob value for logistic regression.

```    # glue constant then apply coeff returned by the logistic method

\$pred = \$x->glue(1,ones(\$x->dim(0)))->pred_logistic( \$m{b} );```
`    my \$d0 = \$y->d0();`

Null deviance for logistic regression.

```    my \$dm = \$y->dm( \$y_pred );

# null deviance
my \$d0 = \$y->dm( ones(\$y->nelem) * \$y->avg );```

Model deviance for logistic regression.

Deviance residual for logistic regression.

## ols_t

Threaded version of ordinary least squares regression (ols). The price of threading was losing significance tests for coefficients (but see r2_change). The fitting function was shamelessly copied then modified from PDL::Fit::Linfit. Uses PDL::Slatec when possible but otherwise uses PDL::MatrixOps. Intercept is LAST of coeff if CONST => 1.

ols_t does not handle bad values. consider fill_m or fill_rand if there are bad values.

Default options (case insensitive):

`    CONST   => 1,`

Usage:

```    # DV, 2 person's ratings for top-10 box office movies
# ascending sorted by box office numbers

perldl> p \$y = qsort ceil( random(10, 2)*5 )
[
[1 1 2 4 4 4 4 5 5 5]
[1 2 2 2 3 3 3 3 5 5]
]

# model with 2 IVs, a linear and a quadratic trend component

perldl> \$x = cat sequence(10), sequence(10)**2

# suppose our novice modeler thinks this creates 3 different models
# for predicting movie ratings

perldl> p \$x = cat \$x, \$x * 2, \$x * 3
[
[
[ 0  1  2  3  4  5  6  7  8  9]
[ 0  1  4  9 16 25 36 49 64 81]
]
[
[  0   2   4   6   8  10  12  14  16  18]
[  0   2   8  18  32  50  72  98 128 162]
]
[
[  0   3   6   9  12  15  18  21  24  27]
[  0   3  12  27  48  75 108 147 192 243]
]
]

perldl> p \$x->info
PDL: Double D [10,2,3]

# insert a dummy dim between IV and the dim (model) to be threaded

perldl> %m = \$y->ols_t( \$x->dummy(2) )

perldl> p "\$_\t\$m{\$_}\n" for (sort keys %m)

# 2 persons' ratings, eached fitted with 3 "different" models

F
[
[ 38.314159  25.087209]
[ 38.314159  25.087209]
[ 38.314159  25.087209]
]

# df is the same across dv and iv models

F_df    [2 7]
F_p
[
[0.00016967051 0.00064215074]
[0.00016967051 0.00064215074]
[0.00016967051 0.00064215074]
]

R2
[
[ 0.9162963 0.87756762]
[ 0.9162963 0.87756762]
[ 0.9162963 0.87756762]
]

b
[
[  0.99015152 -0.056818182   0.66363636]    # person 1
[  0.18939394  0.022727273          1.4]    # person 2
]
[
[  0.49507576 -0.028409091   0.66363636]
[  0.09469697  0.011363636          1.4]
]
[
[  0.33005051 -0.018939394   0.66363636]
[ 0.063131313 0.0075757576          1.4]
]
]

# our novice modeler realizes at this point that
# the 3 models only differ in the scaling of the IV coefficients

ss_model
[
[ 20.616667  13.075758]
[ 20.616667  13.075758]
[ 20.616667  13.075758]
]

ss_residual
[
[ 1.8833333  1.8242424]
[ 1.8833333  1.8242424]
[ 1.8833333  1.8242424]
]

ss_total        [22.5 14.9]
y_pred
[
[
[0.66363636  1.5969697  2.4166667  3.1227273  ...  4.9727273]
...```

## r2_change

Significance test for the incremental change in R2 when new variable(s) are added to an ols regression model. Returns the change stats as well as stats for both models. Based on ols_t. (One way to make up for the lack of significance tests for coeffs in ols_t).

Default options (case insensitive):

`    CONST   => 1,`

Usage:

```    # suppose these are two persons' ratings for top 10 box office movies
# ascending sorted by box office

perldl> p \$y = qsort ceil(random(10, 2) * 5)
[
[1 1 2 2 2 3 4 4 4 4]
[1 2 2 3 3 3 4 4 5 5]
]

# first IV is a simple linear trend

perldl> p \$x1 = sequence 10
[0 1 2 3 4 5 6 7 8 9]

# the modeler wonders if adding a quadratic trend improves the fit

perldl> p \$x2 = sequence(10) ** 2
[0 1 4 9 16 25 36 49 64 81]

# two difference models are given in two pdls
# each as would be pass on to ols_t
# the 1st model includes only linear trend
# the 2nd model includes linear and quadratic trends
# when necessary use dummy dim so both models have the same ndims

perldl> %c = \$y->r2_change( \$x1->dummy(1), cat(\$x1, \$x2) )

perldl> p "\$_\t\$c{\$_}\n" for (sort keys %c)
#              person 1   person 2
F_change        [0.72164948 0.071283096]
# df same for both persons
F_df    [1 7]
F_p     [0.42370145 0.79717232]
R2_change       [0.0085966043 0.00048562549]
model0  HASH(0x8c10828)
model1  HASH(0x8c135c8)

# the answer here is no.```

# METHODS

## anova

Analysis of variance. Uses type III sum of squares for unbalanced data.

Dependent variable should be a 1D pdl. Independent variables can be passed as 1D perl array ref or 1D pdl.

Supports bad value (by ignoring missing or BAD values in dependent and independent variables list-wise).

Default options (case insensitive):

```    V      => 1,          # carps if bad value in variables
IVNM   => [],         # auto filled as ['IV_0', 'IV_1', ... ]
PLOT   => 1,          # plots highest order effect
# can set plot_means options here```

Usage:

```    # suppose this is ratings for 12 apples

perldl> p \$y = qsort ceil( random(12)*5 )
[1 1 2 2 2 3 3 4 4 4 5 5]

# IV for types of apple

perldl> p \$a = sequence(12) % 3 + 1
[1 2 3 1 2 3 1 2 3 1 2 3]

# IV for whether we baked the apple

perldl> @b = qw( y y y y y y n n n n n n )

perldl> %m = \$y->anova( \$a, \@b, { IVNM=>['apple', 'bake'] } )

perldl> p "\$_\t\$m{\$_}\n" for (sort keys %m)
# apple # m
[
[2.5   3 3.5]
]

# apple # se
[
[0.64549722 0.91287093 0.64549722]
]

# apple ~ bake # m
[
[1.5 1.5 2.5]
[3.5 4.5 4.5]
]

# apple ~ bake # se
[
[0.5 0.5 0.5]
[0.5 0.5 0.5]
]

# bake # m
[
[ 1.8333333  4.1666667]
]

# bake # se
[
[0.30731815 0.30731815]
]

F       7.6
F_df    [5 6]
F_p     0.0141586545851857
ms_model        3.8
ms_residual     0.5
ss_model        19
ss_residual     3
ss_total        22
| apple | F     2
| apple | F_df  [2 6]
| apple | F_p   0.216
| apple | ms    1
| apple | ss    2
| apple ~ bake | F      0.666666666666667
| apple ~ bake | F_df   [2 6]
| apple ~ bake | F_p    0.54770848985725
| apple ~ bake | ms     0.333333333333334
| apple ~ bake | ss     0.666666666666667
| bake | F      32.6666666666667
| bake | F_df   [1 6]
| bake | F_p    0.00124263849516693
| bake | ms     16.3333333333333
| bake | ss     16.3333333333333```

## anova_rptd

Repeated measures and mixed model anova. Uses type III sum of squares. The standard error (se) for the means are based on the relevant mean squared error from the anova, ie it is pooled across levels of the effect.

anova_rptd supports bad value in the dependent and independent variables. It automatically removes bad data listwise, ie remove a subject's data if there is any cell missing for the subject.

Default options (case insensitive):

```    V      => 1,    # carps if bad value in dv
IVNM   => [],   # auto filled as ['IV_0', 'IV_1', ... ]
BTWN   => [],   # indices of between-subject IVs (matches IVNM indices)
PLOT   => 1,    # plots highest order effect
# see plot_means() for more options```

Usage:

```    Some fictional data: recall_w_beer_and_wings.txt

Subject Beer    Wings   Recall
Alex    1       1       8
Alex    1       2       9
Alex    1       3       12
Alex    2       1       7
Alex    2       2       9
Alex    2       3       12
Brian   1       1       12
Brian   1       2       13
Brian   1       3       14
Brian   2       1       9
Brian   2       2       8
Brian   2       3       14
...

# rtable allows text only in 1st row and col
my (\$data, \$idv, \$subj) = rtable 'recall_w_beer_and_wings.txt';

my (\$b, \$w, \$dv) = \$data->dog;
# subj and IVs can be 1d pdl or @ ref
# subj must be the first argument
my %m = \$dv->anova_rptd( \$subj, \$b, \$w, {ivnm=>['Beer', 'Wings']} );

print "\$_\t\$m{\$_}\n" for (sort keys %m);

# Beer # m
[
[ 10.916667  8.9166667]
]

# Beer # se
[
[ 0.4614791  0.4614791]
]

# Beer ~ Wings # m
[
[   10     7]
[ 10.5  9.25]
[12.25  10.5]
]

# Beer ~ Wings # se
[
[0.89170561 0.89170561]
[0.89170561 0.89170561]
[0.89170561 0.89170561]
]

# Wings # m
[
[   8.5  9.875 11.375]
]

# Wings # se
[
[0.67571978 0.67571978 0.67571978]
]

ss_residual 19.0833333333333
ss_subject  24.8333333333333
ss_total    133.833333333333
| Beer | F  9.39130434782609
| Beer | F_p        0.0547977008378944
| Beer | df 1
| Beer | ms 24
| Beer | ss 24
| Beer || err df    3
| Beer || err ms    2.55555555555556
| Beer || err ss    7.66666666666667
| Beer ~ Wings | F  0.510917030567687
| Beer ~ Wings | F_p        0.623881438624431
| Beer ~ Wings | df 2
| Beer ~ Wings | ms 1.625
| Beer ~ Wings | ss 3.25000000000001
| Beer ~ Wings || err df    6
| Beer ~ Wings || err ms    3.18055555555555
| Beer ~ Wings || err ss    19.0833333333333
| Wings | F 4.52851711026616
| Wings | F_p       0.0632754786153548
| Wings | df        2
| Wings | ms        16.5416666666667
| Wings | ss        33.0833333333333
| Wings || err df   6
| Wings || err ms   3.65277777777778
| Wings || err ss   21.9166666666667```

For mixed model anova, ie when there are between-subject IVs involved, feed the IVs as above, but specify in BTWN which IVs are between-subject. For example, if we had added age as a between-subject IV in the above example, we would do

```    my %m = \$dv->anova_rptd( \$subj, \$age, \$b, \$w,
{ ivnm=>['Age', 'Beer', 'Wings'], btwn=>[0] });```

## dummy_code

Dummy coding of nominal variable (perl @ ref or 1d pdl) for use in regression.

Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being marked as BAD).

```    perldl> @a = qw(a a a b b b c c c)
perldl> p \$a = dummy_code(\@a)
[
[1 1 1 0 0 0 0 0 0]
[0 0 0 1 1 1 0 0 0]
]```

## effect_code

Unweighted effect coding of nominal variable (perl @ ref or 1d pdl) for use in regression. returns in @ context coded pdl and % ref to level - pdl->dim(1) index.

Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being marked as BAD).

```    my @var = qw( a a a b b b c c c );
my (\$var_e, \$map) = effect_code( \@var );

print \$var_e . \$var_e->info . "\n";

[
[ 1  1  1  0  0  0 -1 -1 -1]
[ 0  0  0  1  1  1 -1 -1 -1]
]
PDL: Double D [9,2]

print "\$_\t\$map->{\$_}\n" for (sort keys %\$map)
a       0
b       1
c       2```

## effect_code_w

Weighted effect code for nominal variable. returns in @ context coded pdl and % ref to level - pdl->dim(1) index.

Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being marked as BAD).

```    perldl> @a = qw( a a b b b c c )
perldl> p \$a = effect_code_w(\@a)
[
[   1    1    0    0    0   -1   -1]
[   0    0    1    1    1 -1.5 -1.5]
]```

## interaction_code

Returns the coded interaction term for effect-coded variables.

Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being marked as BAD).

```    perldl> \$a = sequence(6) > 2
perldl> p \$a = \$a->effect_code
[
[ 1  1  1 -1 -1 -1]
]

perldl> \$b = pdl( qw( 0 1 2 0 1 2 ) )
perldl> p \$b = \$b->effect_code
[
[ 1  0 -1  1  0 -1]
[ 0  1 -1  0  1 -1]
]

perldl> p \$ab = interaction_code( \$a, \$b )
[
[ 1  0 -1 -1 -0  1]
[ 0  1 -1 -0 -1  1]
]```

## ols

Ordinary least squares regression, aka linear regression. Unlike ols_t, ols is not threadable, but it can handle bad value (by ignoring observations with bad value in dependent or independent variables list-wise) and returns the full model in list context with various stats.

IVs (\$x) should be pdl dims \$y->nelem or \$y->nelem x n_iv. Do not supply the constant vector in \$x. Intercept is automatically added and returned as LAST of the coeffs if CONST=>1. Returns full model in list context and coeff in scalar context.

Default options (case insensitive):

```    CONST  => 1,
PLOT   => 1,   # see plot_residuals() for plot options```

Usage:

```    # suppose this is a person's ratings for top 10 box office movies
# ascending sorted by box office

perldl> p \$y = qsort ceil( random(10) * 5 )
[1 1 2 2 2 2 4 4 5 5]

# construct IV with linear and quadratic component

perldl> p \$x = cat sequence(10), sequence(10)**2
[
[ 0  1  2  3  4  5  6  7  8  9]
[ 0  1  4  9 16 25 36 49 64 81]
]

perldl> %m = \$y->ols( \$x )

perldl> p "\$_\t\$m{\$_}\n" for (sort keys %m)

F       40.4225352112676
F_df    [2 7]
F_p     0.000142834216344756
R2      0.920314253647587

b       [0.21212121 0.03030303 0.98181818]
b_p     [0.32800118 0.20303404 0.039910509]
b_se    [0.20174693 0.021579989 0.38987581]
b_t     [ 1.0514223   1.404219  2.5182844]
ss_model        19.8787878787879
ss_residual     1.72121212121212
ss_total        21.6
y_pred  [0.98181818  1.2242424  1.5272727  ...  4.6181818  5.3454545]```

## ols_rptd

Repeated measures linear regression (Lorch & Myers, 1990; Van den Noortgate & Onghena, 2006). Handles purely within-subject design for now. See t/stats_ols_rptd.t for an example using the Lorch and Myers data.

Usage:

```    # This is the example from Lorch and Myers (1990),
# a study on how characteristics of sentences affected reading time
# Three within-subject IVs:
# SP -- serial position of sentence
# WORDS -- number of words in sentence
# NEW -- number of new arguments in sentence

# \$subj can be 1D pdl or @ ref and must be the first argument
# IV can be 1D @ ref or pdl
# 1D @ ref is effect coded internally into pdl
# pdl is left as is

my %r = \$rt->ols_rptd( \$subj, \$sp, \$words, \$new );

print "\$_\t\$r{\$_}\n" for (sort keys %r);

(ss_residual)   58.3754646504336
(ss_subject)    51.8590337714286
(ss_total)  405.188241771429
#      SP        WORDS      NEW
F   [  7.208473  61.354153  1.0243311]
F_p [0.025006181 2.619081e-05 0.33792837]
coeff   [0.33337285 0.45858933 0.15162986]
df  [1 1 1]
df_err  [9 9 9]
ms  [ 18.450705  73.813294 0.57026483]
ms_err  [ 2.5595857  1.2030692 0.55671923]
ss  [ 18.450705  73.813294 0.57026483]
ss_err  [ 23.036272  10.827623  5.0104731]```

## logistic

Logistic regression with maximum likelihood estimation using PDL::Fit::LM (requires PDL::Slatec. Hence loaded with "require" in the sub instead of "use" at the beginning).

IVs (\$x) should be pdl dims \$y->nelem or \$y->nelem x n_iv. Do not supply the constant vector in \$x. It is included in the model and returned as LAST of coeff. Returns full model in list context and coeff in scalar context.

The significance tests are likelihood ratio tests (-2LL deviance) tests. IV significance is tested by comparing deviances between the reduced model (ie with the IV in question removed) and the full model.

***NOTE: the results here are qualitatively similar to but not identical with results from R, because different algorithms are used for the nonlinear parameter fit. Use with discretion***

Default options (case insensitive):

```    INITP => zeroes( \$x->dim(1) + 1 ),    # n_iv + 1
MAXIT => 1000,
EPS   => 1e-7,```

Usage:

```    # suppose this is whether a person had rented 10 movies

perldl> p \$y = ushort( random(10)*2 )
[0 0 0 1 1 0 0 1 1 1]

# IV 1 is box office ranking

perldl> p \$x1 = sequence(10)
[0 1 2 3 4 5 6 7 8 9]

# IV 2 is whether the movie is action- or chick-flick

perldl> p \$x2 = sequence(10) % 2
[0 1 0 1 0 1 0 1 0 1]

# concatenate the IVs together

perldl> p \$x = cat \$x1, \$x2
[
[0 1 2 3 4 5 6 7 8 9]
[0 1 0 1 0 1 0 1 0 1]
]

perldl> %m = \$y->logistic( \$x )

perldl> p "\$_\t\$m{\$_}\n" for (sort keys %m)

D0  13.8629436111989
Dm  9.8627829791575
Dm_chisq    4.00016063204141
Dm_df       2
Dm_p        0.135324414081692
#  ranking    genre      constant
b   [0.41127706 0.53876358 -2.1201285]
b_chisq     [ 3.5974504 0.16835559  2.8577151]
b_p [0.057868258  0.6815774 0.090936587]
iter        12
y_pred      [0.10715577 0.23683909 ... 0.76316091 0.89284423]```

## pca

Principal component analysis. Based on corr instead of cov (bad values are ignored pair-wise. OK when bad values are few but otherwise probably should fill_m etc before pca). Use PDL::Slatec::eigsys() if installed, otherwise use PDL::MatrixOps::eigens_sym().

Default options (case insensitive):

```    CORR  => 1,     # boolean. use correlation or covariance
PLOT  => 1,     # calls plot_screes by default
# can set plot_screes options here```

Usage:

```    my \$d = qsort random 10, 5;      # 10 obs on 5 variables
my %r = \$d->pca( \%opt );
print "\$_\t\$r{\$_}\n" for (keys %r);

eigenvalue    # variance accounted for by each component
[4.70192 0.199604 0.0471421 0.0372981 0.0140346]

eigenvector   # dim var x comp. weights for mapping variables to component
[
[ -0.451251  -0.440696  -0.457628  -0.451491  -0.434618]
[ -0.274551   0.582455   0.131494   0.255261  -0.709168]
[   0.43282   0.500662  -0.139209  -0.735144 -0.0467834]
[  0.693634  -0.428171   0.125114   0.128145  -0.550879]
[  0.229202   0.180393  -0.859217     0.4173  0.0503155]
]

[
[ -0.978489  -0.955601  -0.992316   -0.97901  -0.942421]
[ -0.122661   0.260224  0.0587476   0.114043  -0.316836]
[ 0.0939749   0.108705 -0.0302253  -0.159616 -0.0101577]
[   0.13396 -0.0826915  0.0241629  0.0247483   -0.10639]
[  0.027153  0.0213708  -0.101789  0.0494365 0.00596076]
]

pct_var       # percent variance accounted for by each component
[0.940384 0.0399209 0.00942842 0.00745963 0.00280691]```

Plot scores along the first two components,

`    \$d->plot_scores( \$r{eigenvector} );`

## pca_sorti

Determine by which vars a component is best represented. Descending sort vars by size of association with that component. Returns sorted var and relevant component indices.

Default options (case insensitive):

`    NCOMP => 10,     # maximum number of components to consider`

Usage:

```      # let's see if we replicated the Osgood et al. (1957) study
perldl> (\$data, \$idv, \$ido) = rtable 'osgood_exp.csv', {v=>0}

# select a subset of var to do pca
perldl> \$ind = which_id \$idv, [qw( ACTIVE BASS BRIGHT CALM FAST GOOD HAPPY HARD LARGE HEAVY )]
perldl> \$data = \$data( ,\$ind)->sever
perldl> @\$idv = @\$idv[list \$ind]

perldl> %m = \$data->pca

#   COMP0     COMP1    COMP2    COMP3
HAPPY       [0.860191 0.364911 0.174372 -0.10484]
GOOD        [0.848694 0.303652 0.198378 -0.115177]
CALM        [0.821177 -0.130542 0.396215 -0.125368]
BRIGHT      [0.78303 0.232808 -0.0534081 -0.0528796]
HEAVY       [-0.623036 0.454826 0.50447 0.073007]
HARD        [-0.679179 0.0505568 0.384467 0.165608]
ACTIVE      [-0.161098 0.760778 -0.44893 -0.0888592]
FAST        [-0.196042 0.71479 -0.471355 0.00460276]
LARGE       [-0.241994 0.594644 0.634703 -0.00618055]
BASS        [-0.621213 -0.124918 0.0605367 -0.765184]```

## plot_means

Plots means anova style. Can handle up to 4-way interactions (ie 4D pdl).

Default options (case insensitive):

```    IVNM  => ['IV_0', 'IV_1', 'IV_2', 'IV_3'],
DVNM  => 'DV',
AUTO  => 1,       # auto set dims to be on x-axis, line, panel
# if set 0, dim 0 goes on x-axis, dim 1 as lines
# dim 2+ as panels
# see PDL::Graphics::PGPLOT::Window for next options
WIN   => undef,   # pgwin object. not closed here if passed
# allows comparing multiple lines in same plot
# set env before passing WIN
DEV   => '/xs',         # open and close dev for plotting if no WIN
# defaults to '/png' in Windows
SIZE  => 640,           # individual square panel size in pixels
SYMBL => [0, 4, 7, 11], ```

Usage:

```      # see anova for mean / se pdl structure
\$mean->plot_means( \$se, {IVNM=>['apple', 'bake']} );```

Or like this:

`    \$m{'# apple ~ bake # m'}->plot_means;`

## plot_residuals

Plots residuals against predicted values.

Usage:

`    \$y->plot_residuals( \$y_pred, { dev=>'/png' } );`

Default options (case insensitive):

```     # see PDL::Graphics::PGPLOT::Window for more info
WIN   => undef,  # pgwin object. not closed here if passed
# allows comparing multiple lines in same plot
# set env before passing WIN
DEV   => '/xs',  # open and close dev for plotting if no WIN
# defaults to '/png' in Windows
SIZE  => 640,    # plot size in pixels
COLOR => 1,```

## plot_scores

Plots standardized original and PCA transformed scores against two components. (Thank you, Bob MacCallum, for the documentation suggestion that led to this function.)

Default options (case insensitive):

```  CORR  => 1,      # boolean. PCA was based on correlation or covariance
COMP  => [0,1],  # indices to components to plot
# see PDL::Graphics::PGPLOT::Window for next options
WIN   => undef,  # pgwin object. not closed here if passed
# allows comparing multiple lines in same plot
# set env before passing WIN
DEV   => '/xs',  # open and close dev for plotting if no WIN
# defaults to '/png' in Windows
SIZE  => 640,    # plot size in pixels
COLOR => [1,2],  # color for original and rotated scores```

Usage:

```  my %p = \$data->pca();
\$data->plot_scores( \$p{eigenvector}, \%opt );```

## plot_screes

Scree plot. Plots proportion of variance accounted for by PCA components.

Default options (case insensitive):

```  NCOMP => 20,     # max number of components to plot
CUT   => 0,      # set to plot cutoff line after this many components
# undef to plot suggested cutoff line for NCOMP comps
# see PDL::Graphics::PGPLOT::Window for next options
WIN   => undef,  # pgwin object. not closed here if passed
# allows comparing multiple lines in same plot
# set env before passing WIN
DEV   => '/xs',  # open and close dev for plotting if no WIN
# defaults to '/png' in Windows
SIZE  => 640,    # plot size in pixels
COLOR => 1,```

Usage:

```  # variance should be in descending order

\$pca{var}->plot_screes( {ncomp=>16} );```

Or, because NCOMP is used so often, it is allowed a shortcut,

`  \$pca{var}->plot_screes( 16 );`

PDL::Fit::Linfit

PDL::Fit::LM

# REFERENCES

Cohen, J., Cohen, P., West, S.G., & Aiken, L.S. (2003). Applied Multiple Regression/correlation Analysis for the Behavioral Sciences (3rd ed.). Mahwah, NJ: Lawrence Erlbaum Associates Publishers.

Hosmer, D.W., & Lemeshow, S. (2000). Applied Logistic Regression (2nd ed.). New York, NY: Wiley-Interscience.

Lorch, R.F., & Myers, J.L. (1990). Regression analyses of repeated measures data in cognitive research. Journal of Experimental Psychology: Learning, Memory, & Cognition, 16, 149-157.

Osgood C.E., Suci, G.J., & Tannenbaum, P.H. (1957). The Measurement of Meaning. Champaign, IL: University of Illinois Press.

Rutherford, A. (2001). Introducing Anova and Ancova: A GLM Approach (1st ed.). Thousand Oaks, CA: Sage Publications.

Shlens, J. (2009). A Tutorial on Principal Component Analysis. Retrieved April 10, 2011 from http://citeseerx.ist.psu.edu/

The GLM procedure: unbalanced ANOVA for two-way design with interaction. (2008). SAS/STAT(R) 9.2 User's Guide. Retrieved June 18, 2009 from http://support.sas.com/

Van den Noortgatea, W., & Onghenaa, P. (2006). Analysing repeated measures data in cognitive research: A comment on regression coefficient analyses. European Journal of Cognitive Psychology, 18, 937-952.

# AUTHOR

Copyright (C) 2009 Maggie J. Xiong <maggiexyz users.sourceforge.net>

All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution.

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