Local Moran's I statistic
localmoran.RdThe local spatial statistic Moran's I is calculated for each zone based on the spatial weights object used. The values returned include a Z-value, and may be used as a diagnostic tool. The statistic is: $$I_i = \frac{(x_i-\bar{x})}{{\sum_{k=1}^{n}(x_k-\bar{x})^2}/(n-1)}{\sum_{j=1}^{n}w_{ij}(x_j-\bar{x})}$$, and its expectation and variance were given in Anselin (1995), but those from Sokal et al. (1998) are implemented here.
localmoran(x, listw, zero.policy=attr(listw, "zero.policy"), na.action=na.fail,
conditional=TRUE, alternative = "two.sided", mlvar=TRUE,
spChk=NULL, adjust.x=FALSE)
localmoran_perm(x, listw, nsim=499, zero.policy=attr(listw, "zero.policy"),
na.action=na.fail, alternative = "two.sided", mlvar=TRUE,
spChk=NULL, adjust.x=FALSE, sample_Ei=TRUE, iseed=NULL,
no_repeat_in_row=FALSE)Arguments
- x
a numeric vector the same length as the neighbours list in listw
- listw
a
listwobject created for example bynb2listw- zero.policy
default default
attr(listw, "zero.policy")as set whenlistwwas created, if attribute not set, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA- na.action
a function (default
na.fail), can also bena.omitorna.exclude- in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument tonb2listwmay be subsetted. Ifna.passis used, zero is substituted for NA values in calculating the spatial lag. (Note that na.exclude will only work properly starting from R 1.9.0, na.omit and na.exclude assign the wrong classes in 1.8.*)- conditional
default TRUE: expectation and variance are calculated using the conditional randomization null (Sokal 1998 Eqs. A7 & A8). Elaboration of these changes available in Sauer et al. (2021). If FALSE: expectation and variance are calculated using the total randomization null (Sokal 1998 Eqs. A3 & A4).
- alternative
a character string specifying the alternative hypothesis, must be one of greater, less or two.sided (default).
- mlvar
default TRUE: values of local Moran's I are reported using the variance of the variable of interest (sum of squared deviances over n), but can be reported as the sample variance, dividing by (n-1) instead; both are used in other implementations.
- spChk
should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use
get.spChkOption()- adjust.x
default FALSE, if TRUE, x values of observations with no neighbours are omitted in the mean of x
- nsim
default 499, number of conditonal permutation simulations
- sample_Ei
default TRUE; if conditional permutation, use the sample $E_i$ values, or the analytical values, leaving only variances calculated by simulation.
- iseed
default NULL, used to set the seed for possible parallel RNGs
- no_repeat_in_row
default
FALSE, ifTRUE, sample conditionally in each row without replacements to avoid duplicate values, https://github.com/r-spatial/spdep/issues/124
Details
The values of local Moran's I are divided by the variance (or sample variance) of the variable of interest to accord with Table 1, p. 103, and formula (12), p. 99, in Anselin (1995), rather than his formula (7), p. 98. The variance of the local Moran statistic is taken from Sokal et al. (1998) p. 334, equations 4 & 5 or equations 7 & 8 located depending on user specification. By default, the implementation divides by n, not (n-1) in calculating the variance and higher moments. Conditional code contributed by Jeff Sauer and Levi Wolf.
Value
- Ii
local moran statistic
- E.Ii
expectation of local moran statistic; for
localmoran_permthe permutation sample means- Var.Ii
variance of local moran statistic; for
localmoran_permthe permutation sample standard deviations- Z.Ii
standard deviate of local moran statistic; for
localmoran_permbased on permutation sample means and standard deviations- Pr()
p-value of local moran statistic using
pnorm(); forlocalmoran_permusing standard deviatse based on permutation sample means and standard deviations- Pr() Sim
For
localmoran_perm,rank()andpunif()of observed statistic rank for [0, 1] p-values usingalternative=- Pr(folded) Sim
the simulation folded [0, 0.5] range ranked p-value (based on https://github.com/pysal/esda/blob/4a63e0b5df1e754b17b5f1205b8cadcbecc5e061/esda/crand.py#L211-L213)
- Skewness
For
localmoran_perm, the output ofe1071::skewness()for the permutation samples underlying the standard deviates- Kurtosis
For
localmoran_perm, the output ofe1071::kurtosis()for the permutation samples underlying the standard deviates
In addition, an attribute data frame "quadr" with mean and median quadrant columns, and a column splitting on the demeaned variable and lagged demeaned variable at zero.
Note
Conditional permutations added for comparative purposes; permutations are over the whole data vector omitting the observation itself. For p-value adjustment, use p.adjust() or p.adjustSP() on the output vector.
References
Anselin, L. 1995. Local indicators of spatial association, Geographical Analysis, 27, 93--115; Getis, A. and Ord, J. K. 1996 Local spatial statistics: an overview. In P. Longley and M. Batty (eds) Spatial analysis: modelling in a GIS environment (Cambridge: Geoinformation International), 261--277; Sokal, R. R, Oden, N. L. and Thomson, B. A. 1998. Local Spatial Autocorrelation in a Biological Model. Geographical Analysis, 30. 331--354; Bivand RS, Wong DWS 2018 Comparing implementations of global and local indicators of spatial association. TEST, 27(3), 716--748 doi:10.1007/s11749-018-0599-x ; Sauer, J., Oshan, T. M., Rey, S., & Wolf, L. J. 2021. The Importance of Null Hypotheses: Understanding Differences in Local Moran’s under Heteroskedasticity. Geographical Analysis. doi:10.1111/gean.12304
Bivand, R. (2022), R Packages for Analyzing Spatial Data: A Comparative Case Study with Areal Data. Geographical Analysis, 54(3), 488-518. doi:10.1111/gean.12319
See also
Examples
data(afcon, package="spData")
oid <- order(afcon$id)
resI <- localmoran(afcon$totcon, nb2listw(paper.nb))
printCoefmat(data.frame(resI[oid,], row.names=afcon$name[oid]),
check.names=FALSE)
#> Ii E.Ii Var.Ii Z.Ii
#> THE GAMBIA 3.7523e-01 -2.4317e-02 9.9646e-01 4.0025e-01
#> MALI 4.6363e-01 -2.1841e-02 1.0896e-01 1.4708e+00
#> SENEGAL 2.5670e-01 -3.4444e-03 3.3339e-02 1.4248e+00
#> BENIN 1.9412e-01 -2.4556e-03 2.3792e-02 1.2744e+00
#> MAURITANIA 9.7053e-02 -5.7508e-03 5.5533e-02 4.3625e-01
#> NIGER 2.3071e-01 -1.9459e-02 9.7310e-02 8.0198e-01
#> IVORY COAST 2.9004e-01 -6.9359e-03 5.2072e-02 1.3014e+00
#> GUINEA 1.8263e-01 -2.2246e-03 1.6780e-02 1.4270e+00
#> BURKINA FASO 5.0828e-01 -1.9893e-02 1.1942e-01 1.5284e+00
#> LIBERIA 1.8565e-01 -2.7127e-03 3.5982e-02 9.9300e-01
#> SIERRA LEONE 2.6523e-01 -1.6994e-02 3.4204e-01 4.8257e-01
#> GHANA 1.4764e-01 -1.3414e-03 1.7817e-02 1.1161e+00
#> TOGO 2.1934e-01 -4.9892e-03 6.6025e-02 8.7305e-01
#> CAMEROON 2.5925e-01 -1.1009e-02 8.2313e-02 9.4198e-01
#> NIGERIA 1.1377e-01 -9.6126e-04 9.3272e-03 1.1880e+00
#> GABON 2.0366e-01 -5.4771e-03 1.1153e-01 6.2625e-01
#> CENTRAL AFRICAN REPUBLIC -4.4206e-01 -1.0600e-02 7.9287e-02 -1.5323e+00
#> CHAD -1.0528e-01 -4.0998e-03 2.5008e-02 -6.3985e-01
#> CONGO 1.1380e-02 -8.5953e-04 8.3410e-03 1.3402e-01
#> ZAIRE 7.0978e-01 -5.9545e-02 2.0906e-01 1.6826e+00
#> ANGOLA 1.1797e-01 -6.2140e-04 8.2594e-03 1.3050e+00
#> UGANDA 1.9425e+00 -6.2812e-02 4.4503e-01 3.0060e+00
#> KENYA 1.1969e+00 -1.6803e-02 1.2489e-01 3.4344e+00
#> TANZANIA 2.7185e-01 -4.6254e-02 1.9107e-01 7.2774e-01
#> BURUNDI -4.8428e-01 -1.1009e-02 1.4481e-01 -1.2437e+00
#> RWANDA -7.5236e-01 -1.4730e-02 1.4096e-01 -1.9647e+00
#> SOMALIA 4.5277e-01 -1.1751e-02 2.3778e-01 9.5260e-01
#> ETHIOPIA 7.2512e-01 -5.4929e-03 7.2655e-02 2.7106e+00
#> ZAMBIA 4.2160e-02 -8.1691e-04 3.5354e-03 7.2280e-01
#> ZIMBABWE -9.5068e-03 -6.0969e-03 5.8855e-02 -1.4056e-02
#> MALAWI -2.2888e-01 -1.0284e-02 1.3537e-01 -5.9413e-01
#> MOZAMBIQUE 1.6790e-02 -6.1629e-03 3.7515e-02 1.1850e-01
#> SOUTH AFRICA -1.8254e-01 -5.4306e-03 2.7546e-02 -1.0671e+00
#> LESOTHO -4.1935e-01 -1.9263e-02 7.9348e-01 -4.4914e-01
#> BOTSWANA -3.9316e-03 -1.4141e-04 1.8805e-03 -8.7403e-02
#> SWAZILAND 1.6684e-02 -2.8611e-02 5.6905e-01 6.0045e-02
#> MOROCCO -9.6961e-02 -5.1445e-03 1.0479e-01 -2.8363e-01
#> ALGERIA -1.0037e-02 -9.7828e-05 5.9914e-04 -4.0605e-01
#> TUNISIA 5.3873e-03 -3.0273e-06 6.1985e-05 6.8466e-01
#> LIBYA 8.0382e-01 -1.9923e-02 1.1960e-01 2.3820e+00
#> SUDAN 2.9878e+00 -2.2835e-01 7.6320e-01 3.6814e+00
#> EGYPT 6.9467e+00 -2.9968e-01 4.2971e+00 3.4957e+00
#> Pr.z....E.Ii..
#> THE GAMBIA 0.6890
#> MALI 0.1414
#> SENEGAL 0.1542
#> BENIN 0.2025
#> MAURITANIA 0.6627
#> NIGER 0.4226
#> IVORY COAST 0.1931
#> GUINEA 0.1536
#> BURKINA FASO 0.1264
#> LIBERIA 0.3207
#> SIERRA LEONE 0.6294
#> GHANA 0.2644
#> TOGO 0.3826
#> CAMEROON 0.3462
#> NIGERIA 0.2348
#> GABON 0.5312
#> CENTRAL AFRICAN REPUBLIC 0.1255
#> CHAD 0.5223
#> CONGO 0.8934
#> ZAIRE 0.0925
#> ANGOLA 0.1919
#> UGANDA 0.0026
#> KENYA 0.0006
#> TANZANIA 0.4668
#> BURUNDI 0.2136
#> RWANDA 0.0494
#> SOMALIA 0.3408
#> ETHIOPIA 0.0067
#> ZAMBIA 0.4698
#> ZIMBABWE 0.9888
#> MALAWI 0.5524
#> MOZAMBIQUE 0.9057
#> SOUTH AFRICA 0.2859
#> LESOTHO 0.6533
#> BOTSWANA 0.9304
#> SWAZILAND 0.9521
#> MOROCCO 0.7767
#> ALGERIA 0.6847
#> TUNISIA 0.4936
#> LIBYA 0.0172
#> SUDAN 0.0002
#> EGYPT 0.0005
hist(resI[,5])
mean(resI[,1])
#> [1] 0.4167956
sum(resI[,1])/Szero(nb2listw(paper.nb))
#> [1] 0.4167956
moran.test(afcon$totcon, nb2listw(paper.nb))
#>
#> Moran I test under randomisation
#>
#> data: afcon$totcon
#> weights: nb2listw(paper.nb)
#>
#> Moran I statistic standard deviate = 4.3485, p-value = 6.854e-06
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic Expectation Variance
#> 0.41679563 -0.02439024 0.01029358
#>
# note equality for mean() only when the sum of weights equals
# the number of observations (thanks to Juergen Symanzik)
resI <- localmoran(afcon$totcon, nb2listw(paper.nb))
printCoefmat(data.frame(resI[oid,], row.names=afcon$name[oid]),
check.names=FALSE)
#> Ii E.Ii Var.Ii Z.Ii
#> THE GAMBIA 3.7523e-01 -2.4317e-02 9.9646e-01 4.0025e-01
#> MALI 4.6363e-01 -2.1841e-02 1.0896e-01 1.4708e+00
#> SENEGAL 2.5670e-01 -3.4444e-03 3.3339e-02 1.4248e+00
#> BENIN 1.9412e-01 -2.4556e-03 2.3792e-02 1.2744e+00
#> MAURITANIA 9.7053e-02 -5.7508e-03 5.5533e-02 4.3625e-01
#> NIGER 2.3071e-01 -1.9459e-02 9.7310e-02 8.0198e-01
#> IVORY COAST 2.9004e-01 -6.9359e-03 5.2072e-02 1.3014e+00
#> GUINEA 1.8263e-01 -2.2246e-03 1.6780e-02 1.4270e+00
#> BURKINA FASO 5.0828e-01 -1.9893e-02 1.1942e-01 1.5284e+00
#> LIBERIA 1.8565e-01 -2.7127e-03 3.5982e-02 9.9300e-01
#> SIERRA LEONE 2.6523e-01 -1.6994e-02 3.4204e-01 4.8257e-01
#> GHANA 1.4764e-01 -1.3414e-03 1.7817e-02 1.1161e+00
#> TOGO 2.1934e-01 -4.9892e-03 6.6025e-02 8.7305e-01
#> CAMEROON 2.5925e-01 -1.1009e-02 8.2313e-02 9.4198e-01
#> NIGERIA 1.1377e-01 -9.6126e-04 9.3272e-03 1.1880e+00
#> GABON 2.0366e-01 -5.4771e-03 1.1153e-01 6.2625e-01
#> CENTRAL AFRICAN REPUBLIC -4.4206e-01 -1.0600e-02 7.9287e-02 -1.5323e+00
#> CHAD -1.0528e-01 -4.0998e-03 2.5008e-02 -6.3985e-01
#> CONGO 1.1380e-02 -8.5953e-04 8.3410e-03 1.3402e-01
#> ZAIRE 7.0978e-01 -5.9545e-02 2.0906e-01 1.6826e+00
#> ANGOLA 1.1797e-01 -6.2140e-04 8.2594e-03 1.3050e+00
#> UGANDA 1.9425e+00 -6.2812e-02 4.4503e-01 3.0060e+00
#> KENYA 1.1969e+00 -1.6803e-02 1.2489e-01 3.4344e+00
#> TANZANIA 2.7185e-01 -4.6254e-02 1.9107e-01 7.2774e-01
#> BURUNDI -4.8428e-01 -1.1009e-02 1.4481e-01 -1.2437e+00
#> RWANDA -7.5236e-01 -1.4730e-02 1.4096e-01 -1.9647e+00
#> SOMALIA 4.5277e-01 -1.1751e-02 2.3778e-01 9.5260e-01
#> ETHIOPIA 7.2512e-01 -5.4929e-03 7.2655e-02 2.7106e+00
#> ZAMBIA 4.2160e-02 -8.1691e-04 3.5354e-03 7.2280e-01
#> ZIMBABWE -9.5068e-03 -6.0969e-03 5.8855e-02 -1.4056e-02
#> MALAWI -2.2888e-01 -1.0284e-02 1.3537e-01 -5.9413e-01
#> MOZAMBIQUE 1.6790e-02 -6.1629e-03 3.7515e-02 1.1850e-01
#> SOUTH AFRICA -1.8254e-01 -5.4306e-03 2.7546e-02 -1.0671e+00
#> LESOTHO -4.1935e-01 -1.9263e-02 7.9348e-01 -4.4914e-01
#> BOTSWANA -3.9316e-03 -1.4141e-04 1.8805e-03 -8.7403e-02
#> SWAZILAND 1.6684e-02 -2.8611e-02 5.6905e-01 6.0045e-02
#> MOROCCO -9.6961e-02 -5.1445e-03 1.0479e-01 -2.8363e-01
#> ALGERIA -1.0037e-02 -9.7828e-05 5.9914e-04 -4.0605e-01
#> TUNISIA 5.3873e-03 -3.0273e-06 6.1985e-05 6.8466e-01
#> LIBYA 8.0382e-01 -1.9923e-02 1.1960e-01 2.3820e+00
#> SUDAN 2.9878e+00 -2.2835e-01 7.6320e-01 3.6814e+00
#> EGYPT 6.9467e+00 -2.9968e-01 4.2971e+00 3.4957e+00
#> Pr.z....E.Ii..
#> THE GAMBIA 0.6890
#> MALI 0.1414
#> SENEGAL 0.1542
#> BENIN 0.2025
#> MAURITANIA 0.6627
#> NIGER 0.4226
#> IVORY COAST 0.1931
#> GUINEA 0.1536
#> BURKINA FASO 0.1264
#> LIBERIA 0.3207
#> SIERRA LEONE 0.6294
#> GHANA 0.2644
#> TOGO 0.3826
#> CAMEROON 0.3462
#> NIGERIA 0.2348
#> GABON 0.5312
#> CENTRAL AFRICAN REPUBLIC 0.1255
#> CHAD 0.5223
#> CONGO 0.8934
#> ZAIRE 0.0925
#> ANGOLA 0.1919
#> UGANDA 0.0026
#> KENYA 0.0006
#> TANZANIA 0.4668
#> BURUNDI 0.2136
#> RWANDA 0.0494
#> SOMALIA 0.3408
#> ETHIOPIA 0.0067
#> ZAMBIA 0.4698
#> ZIMBABWE 0.9888
#> MALAWI 0.5524
#> MOZAMBIQUE 0.9057
#> SOUTH AFRICA 0.2859
#> LESOTHO 0.6533
#> BOTSWANA 0.9304
#> SWAZILAND 0.9521
#> MOROCCO 0.7767
#> ALGERIA 0.6847
#> TUNISIA 0.4936
#> LIBYA 0.0172
#> SUDAN 0.0002
#> EGYPT 0.0005
hist(p.adjust(resI[,5], method="bonferroni"))
totcon <-afcon$totcon
is.na(totcon) <- sample(1:length(totcon), 5)
totcon
#> [1] 1363 1421 1861 2355 5246 811 299 358 895 NA NA 933 347 1130 241
#> [16] 604 1015 998 2122 1090 NA NA 758 423 NA 3087 2273 3134 1142 824
#> [31] 2881 487 604 1528 1554 629 792 795 1266 1875 147 363
resI.na <- localmoran(totcon, nb2listw(paper.nb), na.action=na.exclude,
zero.policy=TRUE)
if (class(attr(resI.na, "na.action")) == "exclude") {
print(data.frame(resI.na[oid,], row.names=afcon$name[oid]), digits=2)
} else print(resI.na, digits=2)
#> Ii E.Ii Var.Ii Z.Ii Pr.z....E.Ii..
#> THE GAMBIA 0.3528 -2.9e-02 1.03887 0.37 0.708
#> MALI 0.4524 -2.6e-02 0.11007 1.44 0.149
#> SENEGAL 0.2347 -3.3e-03 0.02800 1.42 0.155
#> BENIN 0.1862 -2.2e-03 0.02572 1.17 0.240
#> MAURITANIA 0.0722 -6.0e-03 0.05043 0.35 0.728
#> NIGER 0.1981 -2.3e-02 0.09763 0.71 0.480
#> IVORY COAST 0.3020 -7.4e-03 0.06218 1.24 0.215
#> GUINEA 0.1774 -2.0e-03 0.01658 1.39 0.164
#> BURKINA FASO 0.5248 -2.3e-02 0.14950 1.42 0.156
#> LIBERIA NA NA NA NA NA
#> SIERRA LEONE 0.2241 -2.0e-02 0.71575 0.29 0.773
#> GHANA 0.1382 -1.0e-03 0.01854 1.02 0.307
#> TOGO NA NA NA NA NA
#> CAMEROON 0.1884 -1.2e-02 0.10306 0.63 0.532
#> NIGERIA 0.0861 -6.6e-04 0.00556 1.16 0.245
#> GABON 0.1828 -5.7e-03 0.10144 0.59 0.554
#> CENTRAL AFRICAN REPUBLIC NA NA NA NA NA
#> CHAD 0.0657 -4.1e-03 0.03430 0.38 0.706
#> CONGO -0.0301 -5.6e-04 0.00652 -0.37 0.715
#> ZAIRE 0.5651 -8.5e-02 0.34207 1.11 0.266
#> ANGOLA 0.2017 -1.5e-03 0.01766 1.53 0.126
#> UGANDA 1.5678 -9.0e-02 0.69207 1.99 0.046
#> KENYA 1.3328 -2.6e-02 0.29012 2.52 0.012
#> TANZANIA 0.4269 -6.7e-02 0.23117 1.03 0.304
#> BURUNDI -0.5611 -1.2e-02 0.14171 -1.46 0.145
#> RWANDA -0.8661 -1.7e-02 0.14069 -2.26 0.024
#> SOMALIA 0.7805 -1.8e-02 0.66673 0.98 0.328
#> ETHIOPIA NA NA NA NA NA
#> ZAMBIA 0.0806 -1.9e-03 0.00690 0.99 0.321
#> ZIMBABWE -0.0393 -6.4e-03 0.05385 -0.14 0.887
#> MALAWI -0.2844 -1.1e-02 0.13166 -0.75 0.452
#> MOZAMBIQUE -0.0121 -6.5e-03 0.03407 -0.03 0.976
#> SOUTH AFRICA -0.2047 -9.1e-03 0.03947 -0.98 0.325
#> LESOTHO -0.5158 -2.3e-02 0.81614 -0.55 0.585
#> BOTSWANA -0.0025 -1.3e-05 0.00015 -0.20 0.838
#> SWAZILAND -0.0494 -3.4e-02 0.59593 -0.02 0.984
#> MOROCCO -0.0936 -8.7e-03 0.15440 -0.22 0.829
#> ALGERIA -0.0143 -4.7e-04 0.00247 -0.28 0.780
#> TUNISIA 0.0428 -1.5e-04 0.00268 0.83 0.406
#> LIBYA 0.5764 -3.0e-02 0.19100 1.39 0.165
#> SUDAN NA NA NA NA NA
#> EGYPT 4.0118 -4.1e-01 8.97211 1.48 0.140
resG <- localG(afcon$totcon, nb2listw(include.self(paper.nb)))
print(data.frame(resG[oid], row.names=afcon$name[oid]), digits=2)
#> resG.oid.
#> THE GAMBIA -0.984
#> MALI -1.699
#> SENEGAL -1.463
#> BENIN -1.301
#> MAURITANIA -0.605
#> NIGER -1.049
#> IVORY COAST -1.417
#> GUINEA -1.449
#> BURKINA FASO -1.751
#> LIBERIA -1.041
#> SIERRA LEONE -0.870
#> GHANA -1.103
#> TOGO -0.991
#> CAMEROON -1.133
#> NIGERIA -1.173
#> GABON -0.789
#> CENTRAL AFRICAN REPUBLIC 1.173
#> CHAD 0.463
#> CONGO -0.203
#> ZAIRE 2.023
#> ANGOLA 1.235
#> UGANDA 3.336
#> KENYA 3.503
#> TANZANIA 1.098
#> BURUNDI 0.774
#> RWANDA 1.457
#> SOMALIA 1.183
#> ETHIOPIA 2.627
#> ZAMBIA 0.753
#> ZIMBABWE -0.200
#> MALAWI 0.212
#> MOZAMBIQUE -0.288
#> SOUTH AFRICA -0.868
#> LESOTHO -0.298
#> BOTSWANA 0.041
#> SWAZILAND -0.659
#> MOROCCO 0.022
#> ALGERIA -0.363
#> TUNISIA 0.579
#> LIBYA 2.553
#> SUDAN 4.039
#> EGYPT 4.421
set.seed(1)
resI_p <- localmoran_perm(afcon$totcon, nb2listw(paper.nb))
printCoefmat(data.frame(resI_p[oid,], row.names=afcon$name[oid]),
check.names=FALSE)
#> Ii E.Ii Var.Ii Z.Ii
#> THE GAMBIA 3.7523e-01 -3.0853e-02 9.0476e-01 4.2692e-01
#> MALI 4.6363e-01 -2.1690e-02 1.2192e-01 1.3899e+00
#> SENEGAL 2.5670e-01 -1.1288e-02 3.2719e-02 1.4816e+00
#> BENIN 1.9412e-01 -6.5012e-03 2.6046e-02 1.2431e+00
#> MAURITANIA 9.7053e-02 -2.0574e-03 5.7209e-02 4.1437e-01
#> NIGER 2.3071e-01 -2.8428e-02 1.1981e-01 7.4867e-01
#> IVORY COAST 2.9004e-01 -8.1011e-03 5.5564e-02 1.2648e+00
#> GUINEA 1.8263e-01 5.3454e-04 1.8397e-02 1.3425e+00
#> BURKINA FASO 5.0828e-01 -2.9727e-02 1.4019e-01 1.4369e+00
#> LIBERIA 1.8565e-01 -1.9247e-03 3.6524e-02 9.8147e-01
#> SIERRA LEONE 2.6523e-01 1.0225e-02 3.1505e-01 4.5432e-01
#> GHANA 1.4764e-01 -6.7639e-03 1.8646e-02 1.1307e+00
#> TOGO 2.1934e-01 -1.8012e-03 6.5187e-02 8.6616e-01
#> CAMEROON 2.5925e-01 -1.0726e-02 9.4789e-02 8.7688e-01
#> NIGERIA 1.1377e-01 -2.5672e-03 8.7720e-03 1.2421e+00
#> GABON 2.0366e-01 -3.5024e-03 1.0958e-01 6.2584e-01
#> CENTRAL AFRICAN REPUBLIC -4.4206e-01 2.3901e-02 8.1700e-02 -1.6302e+00
#> CHAD -1.0528e-01 5.9042e-03 3.0168e-02 -6.4016e-01
#> CONGO 1.1380e-02 7.0349e-03 8.9435e-03 4.5949e-02
#> ZAIRE 7.0978e-01 -2.9589e-02 2.9068e-01 1.3714e+00
#> ANGOLA 1.1797e-01 -3.0421e-04 8.6526e-03 1.2716e+00
#> UGANDA 1.9425e+00 -7.0491e-02 5.2046e-01 2.7903e+00
#> KENYA 1.1969e+00 -1.6362e-02 1.3644e-01 3.2847e+00
#> TANZANIA 2.7185e-01 -7.3309e-02 2.3454e-01 7.1272e-01
#> BURUNDI -4.8428e-01 9.5351e-03 1.4866e-01 -1.2807e+00
#> RWANDA -7.5236e-01 -4.0426e-02 1.7628e-01 -1.6956e+00
#> SOMALIA 4.5277e-01 -1.1165e-02 2.6266e-01 9.0523e-01
#> ETHIOPIA 7.2512e-01 8.3618e-04 7.8477e-02 2.5855e+00
#> ZAMBIA 4.2160e-02 -5.1840e-03 3.6852e-03 7.7989e-01
#> ZIMBABWE -9.5068e-03 -6.5338e-03 6.5357e-02 -1.1629e-02
#> MALAWI -2.2888e-01 3.6276e-03 1.3417e-01 -6.3477e-01
#> MOZAMBIQUE 1.6790e-02 -2.0694e-03 4.5436e-02 8.8476e-02
#> SOUTH AFRICA -1.8254e-01 -1.1611e-02 3.0240e-02 -9.8293e-01
#> LESOTHO -4.1935e-01 -6.6258e-02 8.7314e-01 -3.7787e-01
#> BOTSWANA -3.9316e-03 -4.7713e-03 2.2167e-03 1.7833e-02
#> SWAZILAND 1.6684e-02 -8.4486e-02 6.4807e-01 1.2567e-01
#> MOROCCO -9.6961e-02 -2.1623e-03 1.1253e-01 -2.8260e-01
#> ALGERIA -1.0037e-02 -3.4326e-04 6.3541e-04 -3.8456e-01
#> TUNISIA 5.3873e-03 -1.6456e-04 6.4187e-05 6.9298e-01
#> LIBYA 8.0382e-01 -2.9038e-02 1.3319e-01 2.2821e+00
#> SUDAN 2.9878e+00 -2.5952e-01 8.2522e-01 3.5747e+00
#> EGYPT 6.9467e+00 -3.7869e-01 4.1318e+00 3.6038e+00
#> Pr.z....E.Ii.. Pr.z....E.Ii...Sim Pr.folded..Sim
#> THE GAMBIA 6.6944e-01 8.3200e-01 4.2800e-01
#> MALI 1.6455e-01 8.8000e-02 4.4000e-02
#> SENEGAL 1.3846e-01 6.4000e-02 3.2000e-02
#> BENIN 2.1384e-01 1.3200e-01 6.6000e-02
#> MAURITANIA 6.7860e-01 7.5600e-01 3.7800e-01
#> NIGER 4.5406e-01 5.0000e-01 2.5000e-01
#> IVORY COAST 2.0595e-01 1.1600e-01 5.8000e-02
#> GUINEA 1.7943e-01 7.2000e-02 3.6000e-02
#> BURKINA FASO 1.5075e-01 9.2000e-02 4.6000e-02
#> LIBERIA 3.2636e-01 3.0400e-01 1.5200e-01
#> SIERRA LEONE 6.4960e-01 8.2400e-01 4.1200e-01
#> GHANA 2.5816e-01 1.8400e-01 9.2000e-02
#> TOGO 3.8640e-01 3.6800e-01 1.8400e-01
#> CAMEROON 3.8055e-01 4.3200e-01 2.1600e-01
#> NIGERIA 2.1418e-01 1.4400e-01 7.2000e-02
#> GABON 5.3142e-01 5.8400e-01 2.9600e-01
#> CENTRAL AFRICAN REPUBLIC 1.0306e-01 1.8400e-01 9.2000e-02
#> CHAD 5.2207e-01 4.5200e-01 2.2600e-01
#> CONGO 9.6335e-01 8.5600e-01 4.2800e-01
#> ZAIRE 1.7026e-01 2.0000e-01 1.0000e-01
#> ANGOLA 2.0353e-01 2.3600e-01 1.1800e-01
#> UGANDA 5.2662e-03 4.0000e-02 2.0000e-02
#> KENYA 1.0210e-03 8.0000e-03 4.0000e-03
#> TANZANIA 4.7602e-01 4.2400e-01 2.1200e-01
#> BURUNDI 2.0029e-01 2.6800e-01 1.3400e-01
#> RWANDA 8.9953e-02 1.4000e-01 7.0000e-02
#> SOMALIA 3.6535e-01 2.8800e-01 1.4600e-01
#> ETHIOPIA 9.7244e-03 3.6000e-02 1.8000e-02
#> ZAMBIA 4.3546e-01 4.1200e-01 2.0600e-01
#> ZIMBABWE 9.9072e-01 8.6800e-01 4.3400e-01
#> MALAWI 5.2558e-01 4.5600e-01 2.2800e-01
#> MOZAMBIQUE 9.2950e-01 9.4400e-01 4.7200e-01
#> SOUTH AFRICA 3.2564e-01 3.5200e-01 1.7600e-01
#> LESOTHO 7.0553e-01 4.6800e-01 2.2400e-01
#> BOTSWANA 9.8577e-01 8.4400e-01 4.2200e-01
#> SWAZILAND 8.9999e-01 8.8400e-01 4.4200e-01
#> MOROCCO 7.7749e-01 9.7600e-01 4.8800e-01
#> ALGERIA 7.0057e-01 8.0400e-01 4.0200e-01
#> TUNISIA 4.8832e-01 4.2800e-01 2.1600e-01
#> LIBYA 2.2482e-02 4.8000e-02 2.4000e-02
#> SUDAN 3.5067e-04 8.0000e-03 4.0000e-03
#> EGYPT 3.1358e-04 1.6000e-02 8.0000e-03
#> Skewness Kurtosis
#> THE GAMBIA -1.7643e+00 3.3422
#> MALI -8.2875e-01 0.8397
#> SENEGAL -7.1779e-01 0.3689
#> BENIN -8.5318e-01 0.5520
#> MAURITANIA -9.6301e-01 1.3949
#> NIGER -6.3767e-01 0.2577
#> IVORY COAST -8.3750e-01 0.6786
#> GUINEA -8.8298e-01 0.7691
#> BURKINA FASO -7.7665e-01 0.5911
#> LIBERIA -9.7191e-01 1.2327
#> SIERRA LEONE -1.0737e+00 0.6367
#> GHANA -9.6554e-01 0.8587
#> TOGO -1.0772e+00 1.2283
#> CAMEROON -8.1844e-01 1.1746
#> NIGERIA -6.3061e-01 0.1168
#> GABON -1.1947e+00 1.1977
#> CENTRAL AFRICAN REPUBLIC -6.7964e-01 -0.1428
#> CHAD -9.5123e-01 0.8025
#> CONGO -9.5359e-01 0.9288
#> ZAIRE 6.6929e-01 0.5829
#> ANGOLA 1.0230e+00 1.0841
#> UGANDA 9.6607e-01 1.3331
#> KENYA 8.5205e-01 0.5240
#> TANZANIA 8.5767e-01 1.2127
#> BURUNDI -1.0634e+00 0.9779
#> RWANDA -9.5905e-01 0.8365
#> SOMALIA 1.5185e+00 2.7876
#> ETHIOPIA 9.9587e-01 0.7374
#> ZAMBIA 4.9495e-01 0.1900
#> ZIMBABWE -8.0240e-01 0.4192
#> MALAWI -1.1246e+00 1.7228
#> MOZAMBIQUE -7.6989e-01 0.5793
#> SOUTH AFRICA 5.6394e-01 0.1988
#> LESOTHO -1.7905e+00 2.9865
#> BOTSWANA -9.4637e-01 0.5722
#> SWAZILAND -1.3081e+00 1.7612
#> MOROCCO 1.2705e+00 1.5574
#> ALGERIA 6.3958e-01 0.2677
#> TUNISIA 1.2498e+00 1.2249
#> LIBYA 7.3453e-01 0.5219
#> SUDAN 4.3993e-01 -0.1019
#> EGYPT 1.1875e+00 1.6480