Multivariate Geostatistical Modeling of Phytophthora rubi and Pratylenchus penetrans in Red Raspberry Fields

Exploratory data analysis of disease rating, Phytophthora rubi DNA concentrations in roots, Pratylenchus penetrans densities in roots and soil, soil texture, and elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables	Mean	SE	Standard deviation	VIR^a
OR-1	Disease rating	6.53	0.18	1.37	0.29
	P. rubi DNA (ng/g root)	681.87	192.78	1,493.29	3,270.30
	P. penetrans/g root	16.78	4.80	37.16	82.29
	P. penetrans/g soil	50.38	10.62	82.25	134.27
% sand	27.93	0.50	3.90	0.54
% silt	51.45	0.45	3.50	0.24
% clay	20.61	0.34	2.61	0.33
Elevation (m)	6.43	0.38	2.92	1.33
OR-2	Disease rating	3.65	0.08	0.62	0.10
	P. rubi DNA (ng/g root)	91.75	64.48	499.49	2,719.20
	P. penetrans/g root	337.32	79.88	618.74	1,134.95
	P. penetrans/g soil	86.05	9.06	70.17	57.22
% sand	27.93	0.89	6.87	1.69
% silt	45.97	0.88	6.88	1.03
% clay	26.53	0.44	3.44	0.45
Elevation (m)	0.66	0.05	0.39	0.23
WA-1	Disease rating	3.26	0.23	1.79	0.98
	P. rubi DNA (ng/g root)	1,106.31	514.86	3,988.13	14,376.77
	P. penetrans/g root	16.16	5.96	46.19	132.06
	P. penetrans/g soil	26.77	4.89	37.91	53.67
% sand	62.60	0.89	6.87	0.75
% silt	27.27	0.72	5.57	1.14
% clay	10.13	0.29	2.26	0.50
Elevation (m)	1.74	0.13	1.00	0.58
WA-2	Disease rating	5.47	0.22	1.73	0.54
	P. rubi DNA (ng/g root)	817.32	440.91	3,415.25	14,270.84
	P. penetrans/g root	598.35	97.64	756.33	956.02
	P. penetrans/g soil	178.42	26.01	201.51	227.58
% sand	34.01	1.01	7.85	1.81
% silt	51.75	0.92	7.11	0.98
% clay	14.25	0.41	3.18	0.71
Elevation (m)	1.43	0.09	0.73	0.37

a When <1, the distribution is under-dispersed, and when >1, the distribution is over-dispersed.

SE, standard errors; VMR, variance-to-mean ratio.

Table 2:

Global Moran’s I for assessing spatial autocorrelation in disease rating, Phytophthora rubi DNA concentrations in roots, Pratylenchus penetrans densities in roots and soil, soil texture, and elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables	Moran’s I coefficient^a	P-value^b
OR-1	Disease rating	0.28	<0.001
	P. rubi DNA in root	0.07	0.001
	P. penetrans in root	0.04	0.08
	P. penetrans in soil	0.04	0.07
Sand	0.11	<0.001
Silt	0.10	0.002
Clay	0.18	<0.001
Elevation	0.75	<0.001
OR-2	Disease rating	0.01	0.22
	P. rubi DNA in root	−0.02	0.53
	P. penetrans in root	−0.01	0.40
	P. penetrans in soil	0.06	0.01
Sand	0.07	0.009
Silt	0.06	0.01
Clay	0.02	0.20
Elevation	0.10	0.002
WA-1	Disease rating	0.21	<0.001
	P. rubi DNA in root	0.08	0.003
	P. penetrans in root	0.004	0.29
	P. penetrans in soil	0.31	<0.001
Sand	0.28	<0.001
Silt	0.19	<0.001
Clay	0.31	<0.001
Elevation	0.59	<0.001
WA-2	Disease rating	0.51	<0.001
	P. rubi DNA in root	0.22	<0.001
	P. penetrans in root	0.09	0.003
	P. penetrans in soil	0.11	<0.001
Sand	0.42	<0.001
Silt	0.22	<0.001
Clay	0.43	<0.001
Elevation	0.53	<0.001

a Global Moran’s I coefficient values range from −1 to 1, where a value <0 indicates negative spatial autocorrelation and a value >0 indicates positive spatial autocorrelation. A positive spatial autocorrelation indicates the presence of clustering, i.e., nearby areas have similar attribute values, whereas a negative spatial autocorrelation indicates dissimilarities in neighborhood data attribute values.

b The P-values indicate the level of significance at P < 0.05 for each coefficient.

LISA for disease rating, *Phytophthora rubi* DNA concentrations in roots, and *Pratylenchus penetrans* densities in roots and soil in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). A positive value for the local Moran’s I indicates that a sampling point has a similar attribute value to the neighboring points (clusters), and a negative value indicates dissimilar attribute values in the local neighborhood (outliers). Solid-filled black shapes indicate the significance of the local Moran’s I coefficient at P < 0.05. LISA, local indicators of spatial association.

Moran scatterplot for disease rating, *Phytophthora rubi* DNA concentrations in roots, and *Pratylenchus penetrans* densities in roots and soil in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The scatterplot is characterized by four quadrants corresponding to the four types of spatial association and illustrates relationships between an individual sampling value (standardized observation) and the average values at neighboring sampling points (standardized spatial lags). The lower left and upper right quadrants indicate spatial clustering or PSA. The lower left quadrant is characterized by low values for both the spatial lags and the observations (Low–Low), and in the upper right quadrant, there are high values for both the spatial lags and the observations (High–High). The upper left (Low–High) and lower right (High–Low) quadrants indicate spatial outliers or NSA. The slope of each regression line corresponds to the Moran’s I coefficients listed in Table 2. NSA, negative spatial association; PSA, positive spatial association.

LISA for soil texture (sand, silt, and clay) and field elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). A positive value for the local Moran’s I indicates that a sampling point has a similar attribute value to the neighboring points (clusters), and a negative value indicates dissimilar attribute values in the local neighborhood (outliers). Solid-filled black shapes indicate the significance of the local Moran’s I coefficient at P < 0.05. LISA, local indicators of spatial association.

Moran scatterplot for soil texture (sand, silt, and clay) and field elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The scatterplot is characterized by four quadrants corresponding to the four types of spatial association and illustrates relationships between an individual sampling value (standardized observation) and the average values at neighboring sampling points (standardized spatial lags). The lower left and upper right quadrants indicate spatial clustering or PSA. The lower left quadrant is characterized by low values for both the spatial lags and the observations (Low–Low), and in the upper right quadrant, there are high values for both the spatial lags and the observations (High–High). The upper left (Low–High) and lower right (High–Low) quadrants indicate spatial outliers or NSA. The slope of each regression line corresponds to the Moran’s I coefficients listed in Table 2. NSA, negative spatial association; PSA, positive spatial association.

Experimental variogram graph for assessing the spatial structure of disease rating, *Phytophthora rubi* DNA concentrations in roots, and *Pratylenchus penetrans* densities in roots and soil in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The x-axis shows the distance lag, which is created by plotting the average variability between pairs of sampling points, and the y-axis shows the calculated value of the semivariance, where a greater value indicates less spatial correlation between pairs of points. The variogram model type and fitting parameters for each commercial field are listed in Table 3.

Experimental variogram graph for assessing the spatial structure of soil texture (sand, silt, and clay) and field elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The x-axis shows the distance lag, which is created by plotting the average variability between pairs of sampling points, and the y-axis shows the calculated value of the semivariance, where a greater value indicates less spatial correlation between pairs of points. The variogram model type and fitting parameters for each commercial field are listed in Table 3.

Table 3:

Fitted models for the experimental variogram analysis for quantifying the extent of spatial autocorrelation in disease rating, Phytophthora rubi DNA concentrations in roots, Pratylenchus penetrans densities in roots and soil, soil texture, and elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables	Model^a	Nugget^b	Sill^c	Range^d
OR-1	Disease rating	Gaussian	0.01	0.02	102.27
	P. rubi DNA in root	Gaussian	1.05	0.44	100.72
	P. penetrans in root	Gaussian	0.50	0.03	29.52
	P. penetrans in soil	Gaussian	0.76	0.11	50.01
Sand	Exponential	0.002	0.003	625.91
Silt	Exponential	0.001	0.003	983.76
Clay	Gaussian	0.001	0.0005	73.70
Elevation	Gaussian	0.002	0.41	107.19
OR-2	Disease rating	Exponential	0.004	0.004	644.08
	P. rubi DNA in root	Gaussian	0.54	0.03	36.47
	P. penetrans in root	Exponential	1.62	1.17	674.35
	P. penetrans in soil	Exponential	0	0.39	10.03
Sand	Gaussian	0.004	0.002	72.89
Silt	Exponential	0.003	0.004	105.97
Clay	Gaussian	0.001	0.0007	56.90
Elevation	Exponential	0.044	0.15	861.29
WA-1	Disease rating	Gaussian	0.03	0.32	439.20
	P. rubi DNA in root	Exponential	0.37	1.98	37.66
	P. penetrans in root	Spherical	0.31	0.14	48.26
	P. penetrans in soil	Spherical	0.08	0.56	94.31
Sand	Spherical	0.003	0.004	505.74
Silt	Gaussian	0.003	0.002	75.60
Clay	Exponential	0.001	0.0004	145.25
Elevation	Gaussian	0.024	104.36	2,855.12
WA-2	Disease rating	Exponential	0.0005	0.09	265.02
	P. rubi DNA in root	Exponential	0.89	0.89	362.76
	P. penetrans in root	Exponential	0.62	0.74	23.00
	P. penetrans in soil	Exponential	0.13	0.42	49.90
Sand	Gaussian	0.002	0.004	59.61
Silt	Exponential	0.002	0.006	82.70
Clay	Exponential	0	0.002	79.82
Elevation	Gaussian	0.01	0.10	96.70

a The fitted mathematical model for the experimental variogram.

b The short-range variability in the data.

c The total variance of the empirical variogram.

d The distance after which the data are no longer correlated.

Directional experimental variogram graph for assessing potential directional variations in the spatial structure of disease rating, *Phytophthora rubi* DNA concentrations in roots, and *Pratylenchus penetrans* densities in roots and soil in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). Directional angles used for the graph include 0° (North–South), 45° (Northeast–Southwest), 90° (East–West), and 135° (Southeast–Northwest). The overlapping of all the angle point lines indicates no directional variations in the spatial structure of the attribute values (isotropy), while major gaps between the angle point lines indicate the presence of directional trends (anisotropy).

Directional experimental variogram graph for assessing potential directional variations in the spatial structure of soil texture (sand, silt, and clay) and field elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). Directional angles used for the graph include 0° (North–South), 45° (Northeast–Southwest), 90° (East–West), and 135° (Southeast–Northwest). The overlapping of all the angle point lines indicates no directional variations in the spatial structure of the attribute values (isotropy), while major gaps between the angle point lines indicate the presence of directional trends (anisotropy).

Kriging density graph for comparing the distribution of interpolated values of disease rating, *Phytophthora rubi* DNA concentrations in roots, and *Pratylenchus penetrans* densities in roots and soil across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The x-axis shows the attribute values of the variable, and the y-axis shows the proportion value assigned to each range of the attribute values.

Kriging density graph for comparing the distribution of interpolated values of soil texture (sand, silt, and clay) and field elevation across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The x-axis shows the attribute values of the variable, and the y-axis shows the proportion value assigned to each range of the attribute values.

Spatial distribution of disease rating across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict disease ratings at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of *Phytophthora rubi* DNA concentrations in roots across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict *P. rubi* DNA in roots at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of *Pratylenchus penetrans* densities in roots across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict *P. penetrans* densities in roots at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of *Pratylenchus penetrans* densities in soil across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict *P. penetrans* in the soil at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of the percentage of sand across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict the percentage of sand at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of the percentage of silt across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict the percentage of silt at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of the percentage of clay across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict the percentage of clay at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Spatial distribution of field elevation across four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2). The ordinary kriging was used as a probabilistic interpolator to predict field elevation at one or more unsampled points. Northing divides the map from north to south and Easting from west to east. Northing and Easting are grid references and are expressed in meters.

Table 4:

Coefficient estimates and SE for the OLS, SLM, SDM, SEM, CAR, and SARMA multiple regression models and associated VIF and P values for Phytophthora rubi DNA concentrations in roots regressed against soil texture (sand, silt, and clay) and elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables^a	OLS					SLM				SDM				SEM				CAR				SARMA
Field	Variables^a	Estimate	SE	P	VIF	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model
OR-1	Intercept	170.82	176.79	0.34	…	…	162.61	168.89	0.34	…	630.02	1,137.13	0.58	…	206.50	171.30	0.23	…	165.21	169.63	0.33	…	352.59	174.37	0.04	…
	Sand	−90.38	93.71	0.34	834	…	−86.08	89.52	0.34	…	−58.62	85.97	0.50	…	−109.83	90.71	0.23	…	−87.16	89.91	0.33	…	−190.22	91.85	0.04	…
	Silt	−106.57	106.04	0.32	672	…	−101.50	101.30	0.32	….	−69.18	97.38	0.48	….	−129.10	102.67	0.21	….	−103.13	101.73	0.31	….	−219.04	104.35	0.04	….
	Clay	−71.96	84.94	0.40	371	….	−68.82	81.13	0.40	….	−43.99	78.40	0.57	….	−86.67	82.50	0.29	….	−69.55	81.53	0.39	….	−150.11	84.75	0.08	….
	Elevation	0.14	0.21	0.52	1	….	0.14	0.21	0.51	….	−0.27	0.56	0.63	….	0.18	0.16	0.26	….	0.12	0.20	0.55	….	0.34	0.07	<0.001	….
P	….	….	….	….	0.12	….	….	….	0.61	….	….	….	0.002	….	….	….	0.52	….	….	….	0.81	….	….	….	<0.001
R²	….	….	….	….	0.12	….	….	….	0.13	….	….	….	0.31	….	….	….	0.13	….	….	….	0.12	….	….	….	0.27
ρ	….	….	….	….	….	….	….	….	0.16	….	….	….	−1.61	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.45	….	….	….	−0.11	….	….	….	−1.00
AIC	….	….	….	….	188.75	….	….	….	190.52	….	….	….	184.66	….	….	….	190.33	….	….	….	190.69	….	….	….	179.81
OR-2	Intercept	3.57	5.96	0.55	….	….	3.77	5.69	0.51	….	64.68	34.00	0.06	….	3.71	5.71	0.52	….	3.81	5.71	0.50	….	3.75	5.71	0.51	….
	Sand	−3.33	3.36	0.33	7	….	−3.35	3.21	0.30	….	−4.27	2.96	0.15	….	−3.29	3.21	0.30	….	−3.31	3.21	0.30	….	−3.25	3.20	0.31	….
	Silt	−1.35	2.73	0.62	4	….	−1.39	2.61	0.59	….	−0.90	2.37	0.71	….	−1.35	2.62	0.61	….	−1.33	2.62	0.61	….	−1.33	2.62	0.61	….
	Clay	−1.74	4.84	0.72	4	….	−1.95	4.63	0.67	….	−4.48	4.31	0.30	….	−2.11	4.64	0.65	….	−2.24	4.64	0.63	….	−2.28	4.65	0.62	….
	Elevation	0.70	0.43	0.11	1	….	0.72	0.41	0.08	….	0.99	0.43	0.02	….	0.74	0.41	0.07	….	0.71	0.41	0.08	….	0.77	0.41	0.06	….
P	….	….	….	….	0.20	….	….	….	0.66	….	….	….	0.08	….	….	….	0.75	….	….	….	0.80	….	….	….	0.70
R²	….	….	….	….	0.10	….	….	….	0.10	….	….	….	0.26	….	….	….	0.10	….	….	….	0.10	….	….	….	0.10
ρ	….	….	….	….	….	….	….	….	−0.19	….	….	….	−0.95	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.16	….	….	….	−0.21	….	….	….	−0.22
AIC	….	….	….	….	140.22	….	….	….	142.03	….	….	….	138.41	….	….	….	142.12	….	….	….	142.16	….	….	….	142.07
WA-1	Intercept	377.42	178.96	0.04	….	….	366.48	171.24	0.03	….	2,253.52	928.36	0.02	….	433.58	172.67	0.01	….	394.63	172.45	0.02	….	623.12	168.46	<0.001	….
	Sand	−234.25	110.15	0.04	2,273	….	−227.56	105.39	0.03	….	−307.03	109.63	0.01	….	−269.09	106.27	0.01	….	−245.12	106.14	0.02	….	−385.75	103.65	<0.001	….
	Silt	−216.77	102.23	0.04	1,513	….	−210.62	97.81	0.03	….	−283.21	101.50	0.01	….	−249.03	98.70	0.01	….	−226.96	98.56	0.02	….	−359.67	96.37	<0.001	….
	Clay	−136.93	69.69	0.05	234	….	−132.92	66.68	0.05	….	−182.98	69.40	0.01	….	−157.58	67.22	0.02	….	−142.08	67.14	0.03	….	−226.26	65.59	<0.001	….
	Elevation	0.16	0.59	0.79	2	….	0.23	0.59	0.70	….	0.61	0.96	0.53	….	0.16	0.53	0.76	….	0.14	0.54	0.80	….	0.03	0.47	0.95	….
P	….	….	….	….	0.06	….	….	….	0.79	….	….	….	0.32	….	….	….	0.56	….	….	….	0.64	….	….	….	<0.001
R²	….	….	….	….	0.15	….	….	….	0.15	….	….	….	0.26	….	….	….	0.16	….	….	….	0.16	….	….	….	0.33
ρ	….	….	….	….	….	….	….	….	0.10	….	….	….	−0.56	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.35	….	….	….	−0.47	….	….	….	−1.00
AIC	….	….	….	….	205.23	….	….	….	207.16	….	….	….	207.30	….	….	….	206.89	….	….	….	207.01	….	….	….	192.91
WA-2	Intercept	31.13	81.32	0.70	….	….	55.75	74.27	0.45	….	−1,283.90	435.02	0.003	….	81.42	72.40	0.26	….	73.91	75.04	0.32	….	81.12	68.18	0.23	….
	Sand	−13.68	46.87	0.77	918	….	−29.74	42.81	0.49	….	1.43	45.08	0.97	….	−44.84	41.77	0.28	….	−39.77	43.26	0.36	….	−45.34	39.32	0.25	….
	Silt	−20.09	48.30	0.68	740	….	−35.34	44.11	0.42	….	−3.13	46.18	0.95	….	−50.61	43.02	0.24	….	−46.71	44.59	0.29	….	−50.56	40.54	0.21	….
	Clay	−14.28	35.07	0.69	150	….	−22.02	32.03	0.49	….	7.21	33.75	0.83	….	−31.23	31.03	0.31	….	−29.04	32.27	0.37	….	−29.98	29.13	0.30	….
	Elevation	0.22	0.41	0.60	1	….	0.07	0.38	0.86	….	0.36	0.66	0.58	….	0.21	0.55	0.71	….	0.47	0.49	0.34	….	0.30	0.59	0.61	….
P	….	….	….	….	0.01	….	….	….	0.04	….	….	….	0.72	….	….	….	0.05	….	….	….	0.16	….	….	….	0.01
R²	….	….	….	….	0.21	….	….	….	0.26	….	….	….	0.39	….	….	….	0.26	….	….	….	0.24	….	….	….	0.29
ρ	….	….	….	….	….	….	….	….	0.56	….	….	….	−0.17	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.70	….	….	….	0.76	….	….	….	1.82
AIC	….	….	….	….	174.41	….	….	….	172.22	….	….	….	168.70	….	….	….	172.43	….	….	….	174.42	….	….	….	170.42

a Independent variables were sand, silt, clay, and field elevation. The P value represents the significance of the model at P < 0.05. R² is the Nagelkerke pseudo-R-squared; ρ is a fitting parameter for SLM and SDM; λ is a fitting parameter for SEM, CAR, and SARMA. AIC was used to compare OLS, SLM, SDM, SEM, CAR, and SARMA models and determine the best model that fits the data for each field. The lower the AIC, the better the model.

AIC, Akaike information criterion; CAR, conditional autoregressive; OLS, ordinary least squares; P, probability; SARMA, spatial autoregressive moving average; SDM, spatial Durbin model; SE, standard errors; SEM, spatial error model; SLM, spatial lag model; VIF, variance inflation factor.

Table 5:

Coefficient estimates and SE for the OLS, SLM, SDM, SEM, CAR, and SARMA multiple regression models and associated VIF and P values for Pratylenchus penetrans densites in roots regressed against soil texture (sand, silt, and clay) and elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables^a	OLS					SLM				SDM				SEM				CAR				SARMA
Field	Variables^a	Estimate	SE	P	VIF	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model
OR-1	Intercept	22.66	119.13	0.85	….	….	22.73	114.06	0.84	….	973.21	805.65	0.23	….	29.70	115.29	0.80	….	28.07	114.95	0.80	….	−0.03	111.84	0.99	….
	Sand	−12.03	63.15	0.85	834	….	−12.05	60.46	0.84	….	−57.46	62.92	0.36	….	−15.45	61.03	0.80	….	−14.49	60.60	0.81	….	1.24	58.91	0.98	….
	Silt	−10.88	71.46	0.88	672	….	−10.86	68.42	0.87	….	−62.92	71.27	0.38	….	−14.67	69.08	0.83	….	−13.61	68.92	0.84	….	3.39	66.93	0.96	….
	Clay	−14.89	57.24	0.80	371	….	−15.03	54.80	0.78	….	−55.89	57.40	0.33	….	−19.50	55.57	0.73	….	−18.83	55.31	0.73	….	−6.89	54.35	0.90	….
	Elevation	0.10	0.14	0.50	1	….	0.11	0.14	0.44	….	−0.34	0.41	0.42	….	0.12	0.10	0.22	….	0.09	0.12	0.46	….	0.15	0.05	0.002	….
P	….	….	….	….	0.37	….	….	….	0.87	….	….	….	0.14	….	….	….	0.35	….	….	….	0.52	….	….	….	<0.001
R²	….	….	….	….	0.07	….	….	….	0.07	….	….	….	0.18	….	….	….	0.09	….	….	….	0.08	….	….	….	0.31
ρ	….	….	….	….	….	….	….	….	−0.07	….	….	….	−0.88	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.60	….	….	….	−0.58	….	….	….	−0.99
AIC	….	….	….	….	141.38	….	….	….	143.35	….	….	….	143.80	….	….	….	142.50	….	….	….	142.96	….	….	….	125.53
OR-2	Intercept	3.49	10.82	0.75	….	….	3.60	10.36	0.73	….	25.86	63.55	0.68	….	3.27	10.36	0.75	….	3.77	10.36	0.72	….	3.34	10.35	0.75	….
	Sand	0.35	6.10	0.95	7	….	0.44	5.84	0.94	….	−1.02	5.62	0.86	….	0.83	5.81	0.89	….	0.60	5.83	0.92	….	1.23	5.76	0.83	….
	Silt	−3.42	4.96	0.49	4	….	−3.41	4.74	0.47	….	−3.60	4.49	0.42	….	−3.35	4.76	0.48	….	−3.38	4.75	0.48	….	−3.45	4.77	0.47	….
	Clay	1.80	8.80	0.84	4	….	1.71	8.42	0.84	….	−0.04	8.11	0.99	….	1.55	8.43	0.85	….	0.99	8.43	0.91	….	1.10	8.46	0.90	….
	Elevation	−0.92	0.79	0.25	1	….	−0.90	0.75	0.23	….	−1.18	0.80	0.14	….	−0.88	0.74	0.24	….	−0.94	0.75	0.21	….	−0.88	0.72	0.22	….
P	….	….	….	….	0.49	….	….	….	0.80	….	….	….	0.19	….	….	….	0.66	….	….	….	0.77	….	….	….	0.51
R²	….	….	….	….	0.06	….	….	….	0.06	….	….	….	0.17	….	….	….	0.06	….	….	….	0.06	….	….	….	0.07
ρ	….	….	….	….	….	….	….	….	−0.10	….	….	….	−0.64	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.22	….	….	….	−0.18	….	….	….	−0.43
AIC	….	….	….	….	211.88	….	….	….	213.82	….	….	….	214.17	….	….	….	213.68	….	….	….	213.80	….	….	….	213.44
WA-1	Intercept	−76.26	100.24	0.45	….	….	−77.12	95.99	0.42	….	179.96	501.46	0.72	….	−76.26	96.02	0.43	….	−76.81	96.01	0.42	….	−76.27	96.05	0.43	….
	Sand	46.61	61.70	0.45	2,272	….	47.17	59.08	0.42	….	36.20	62.66	0.56	….	46.62	59.10	0.43	….	46.94	59.09	0.43	….	46.62	59.12	0.43	….
	Silt	44.13	57.26	0.44	1,513	….	44.59	54.83	0.42	….	33.18	58.03	0.57	….	44.12	54.85	0.42	….	44.43	54.84	0.42	….	44.11	54.87	0.42	….
	Clay	29.62	39.04	0.45	234	….	30.00	37.38	0.42	….	22.51	39.73	0.57	….	29.63	37.39	0.43	….	29.86	37.39	0.42	….	29.64	37.41	0.43	….
	Elevation	0.39	0.33	0.24	2	….	0.40	0.32	0.22	….	0.23	0.56	0.68	….	0.39	0.32	0.21	….	0.39	0.32	0.21	….	0.39	0.32	0.21	….
P	….	….	….	….	0.58	….	….	….	0.88	….	….	….	0.57	….	….	….	0.98	….	….	….	0.99	….	….	….	0.97
R²	….	….	….	….	0.05	….	….	….	0.05	….	….	….	0.12	….	….	….	0.05	….	….	….	0.05	….	….	….	0.05
ρ	….	….	….	….	….	….	….	….	−0.07	….	….	….	−0.25	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.01	….	….	….	−0.02	….	….	….	−0.02
AIC	….	….	….	….	135.68	….	….	….	137.66	….	….	….	141.14	….	….	….	137.68	….	….	….	137.68	….	….	….	137.68
WA-2	Intercept	−69.62	92.71	0.46	….	….	−80.33	88.63	0.36	….	−555.77	494.67	0.26	….	−85.09	83.81	0.31	….	−105.32	88.00	0.23	….	−102.97	81.30	0.20	….
	Sand	42.80	53.43	0.43	918	….	49.75	51.18	0.33	….	49.42	51.62	0.34	….	52.09	48.36	0.28	….	63.01	50.72	0.21	….	62.42	46.93	0.18	….
	Silt	43.95	55.07	0.43	740	….	50.55	52.68	0.34	….	52.80	52.88	0.32	….	52.72	49.63	0.29	….	65.84	52.22	0.21	….	63.45	48.10	0.19	….
	Clay	23.00	39.98	0.57	150	….	26.83	38.16	0.48	….	31.98	38.73	0.41	….	29.20	36.41	0.42	….	37.54	38.05	0.32	….	36.09	35.40	0.31	….
	Elevation	0.81	0.47	0.09	1	….	1	0.47	0.03	….	0.24	0.77	0.75	….	1.04	0.26	<0.001	….	0.81	0.37	0.03	….	1.19	0.17	<0.001	….
P	….	….	….	….	0.03	….	….	….	0.45	….	….	….	0.02	….	….	….	0.04	….	….	….	0.23	….	….	….	<0.001
R²	….	….	….	….	0.18	….	….	….	0.19	….	….	….	0.29	….	….	….	0.24	….	….	….	0.20	….	….	….	0.42
ρ	….	….	….	….	….	….	….	….	−0.29	….	….	….	−1.29	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−1.25	….	….	….	−0.87	….	….	….	−0.99
AIC	….	….	….	….	190.14	….	….	….	191.57	….	….	….	191.06	….	….	….	187.75	….	….	….	190.73	….	….	….	170.95

Table 6:

Coefficient estimates and SE for the OLS, SLM, SDM, SEM, CAR, and SARMA multiple regression models and associated VIF and P values for Pratylenchus penetrans in soil regressed against soil texture (sand, silt, and clay) and elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables^a	OLS					SLM				SDM				SEM				CAR				SARMA
Field	Variables^a	Estimate	SE	P	VIF	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model
OR-1	Intercept	−36.41	148.53	0.80	….	….	−37.35	142.21	0.79	….	1,044.05	848.43	0.22	….	−68.40	143.65	0.63	….	−52.73	143.26	0.71	….	−71.80	144.06	0.62	….
	Sand	19.96	78.73	0.80	834	….	20.50	75.38	0.78	….	−70.87	63.18	0.26	….	38.28	76.03	0.61	….	29.76	75.90	0.69	….	39.88	76.26	0.60	….
	Silt	25.04	89.09	0.78	672	….	25.66	85.30	0.76	….	−78.09	71.53	0.27	….	45.24	86.07	0.60	….	35.89	85.89	0.68	….	47.07	86.33	0.59	….
	Clay	12.55	71.36	0.86	371	….	12.91	68.32	0.85	….	−70.68	57.59	0.22	….	24.91	69.27	0.72	….	17.65	68.94	0.80	….	27.04	69.43	0.70	….
	Elevation	0.13	0.18	0.48	1	….	0.13	0.18	0.46	….	0.82	0.42	0.05	….	0.06	0.12	0.61	….	0.04	0.15	0.77	….	0.08	0.12	0.53	….
P	….	….	….	….	0.36	….	….	….	0.92	….	….	….	<0.001	….	….	….	0.31	….	….	….	0.48	….	….	….	0.37
R²	….	….	….	….	0.07	….	….	….	0.07	….	….	….	0.42	….	….	….	0.09	….	….	….	0.08	….	….	….	0.09
ρ	….	….	….	….	….	….	….	….	−0.03	….	….	….	−1.95	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.67	….	….	….	−0.60	….	….	….	−0.43
AIC	….	….	….	….	167.85	….	….	….	169.84	….	….	….	149.76	….	….	….	168.8	….	….	….	169.35	….	….	….	169.04
OR-2	Intercept	0.34	5.17	0.95	….	….	−0.45	4.92	0.93	….	15.33	31.00	0.62	….	−0.02	4.87	0.99	….	0.03	4.94	0.99	….	−0.01	4.88	0.99	….
	Sand	0.76	2.92	0.80	7	….	0.87	2.76	0.75	….	−0.32	2.74	0.91	….	0.82	2.77	0.77	….	0.69	2.79	0.80	….	0.79	2.77	0.78	….
	Silt	−0.36	2.37	0.88	4	….	−0.15	2.24	0.94	….	−1.28	2.21	0.56	….	−0.02	2.22	0.99	….	−0.34	2.26	0.88	….	−0.01	2.23	0.99	….
	Clay	3.04	4.20	0.47	4	….	3.13	3.98	0.43	….	1.99	3.95	0.61	….	3.10	3.96	0.43	….	3.60	4.01	0.37	….	3.13	3.97	0.43	….
	Elevation	−0.53	0.38	0.16	1	….	−0.49	0.36	0.16	….	−0.84	0.40	0.03	….	−0.49	0.36	0.18	….	−0.49	0.36	0.18	….	−0.50	0.36	0.17	….
P	….	….	….	….	0.64	….	….	….	0.32	….	….	….	0.95	….	….	….	0.35	….	….	….	0.69	….	….	….	0.37
R²	….	….	….	….	0.04	….	….	….	0.06	….	….	….	0.14	….	….	….	0.06	….	….	….	0.05	….	….	….	0.06
ρ	….	….	….	….	….	….	….	….	0.29	….	….	….	−0.03	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.30	….	….	….	0.11	….	….	….	0.34
AIC	….	….	….	….	123.25	….	….	….	124.26	….	….	….	126.93	….	….	….	124.40	….	….	….	125.09	….	….	….	124.45
WA-1	Intercept	30.66	99.00	0.76	….	….	13.45	86.71	0.88	….	2,135.04	424.48	<0.001	….	−33.96	86.30	0.69	….	−4.70	90.26	0.96	….	−58.12	84.84	0.49	….
	Sand	−17.59	60.93	0.77	2,273	….	−7.27	53.37	0.89	….	−140.67	52.15	0.007	….	21.97	53.10	0.68	….	4.29	55.54	0.93	….	37.00	52.21	0.48	….
	Silt	−17.61	56.55	0.76	1,513	….	−7.56	49.53	0.88	….	−132.37	48.37	0.006	….	19.72	49.16	0.69	….	2.66	51.48	0.96	….	33.30	48.34	0.49	….
	Clay	−15.32	38.55	0.69	234	….	−8.53	33.77	0.80	….	−93.85	33.09	0.004	….	10.04	33.65	0.77	….	−2.33	35.17	0.95	….	19.54	33.11	0.55	….
	Elevation	0.68	0.33	0.04	2	….	0.22	0.30	0.47	….	−0.63	0.44	0.15	….	0.38	0.39	0.33	….	0.49	0.35	0.16	….	0.45	0.37	0.22	….
P	….	….	….	….	<0.001	….	….	….	0.004	….	….	….	0.96	….	….	….	0.02	….	….	….	0.07	….	….	….	0.02
R²	….	….	….	….	0.29	….	….	….	0.38	….	….	….	0.59	….	….	….	0.35	….	….	….	0.32	….	….	….	0.35
ρ	….	….	….	….	….	….	….	….	0.70	….	….	….	−0.02	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.72	….	….	….	0.77	….	….	….	1.01
AIC	….	….	….	….	134.18	….	….	….	127.78	….	….	….	111.71	….	….	….	130.76	….	….	….	133.03	….	….	….	130.58
WA-2	Intercept	−60.06	53.39	0.26	….	….	−51.53	49.99	0.30	….	157.38	292.91	0.59	….	−52.72	49.96	0.29	….	−55.47	50.96	0.28	….	−40.27	49.21	0.41	….
	Sand	36.21	30.77	0.24	918	….	30.52	28.82	0.29	….	5.95	30.84	0.85	….	31.58	28.80	0.27	….	33.65	29.37	0.25	….	24.24	28.37	0.39	….
	Silt	38.69	31.71	0.23	740	….	32.91	29.70	0.27	….	9.24	31.60	0.77	….	34.22	29.69	0.25	….	35.80	30.27	0.24	….	26.65	29.25	0.36	….
	Clay	22.07	23.02	0.34	150	….	18.93	21.55	0.38	….	3.47	23.16	0.88	….	20.09	21.48	0.35	….	20.22	21.96	0.36	….	15.37	21.12	0.47	….
	Elevation	−0.03	0.27	0.92	1	….	−0.04	0.25	0.87	….	−0.39	0.46	0.39	….	−0.11	0.32	0.73	….	0.003	0.27	0.99	….	−0.12	0.34	0.73	….
P	….	….	….	….	0.08	….	….	….	0.17	….	….	….	0.74	….	….	….	0.38	….	….	….	0.71	….	….	….	0.24
R²	….	….	….	….	0.15	….	….	….	0.17	….	….	….	0.25	….	….	….	0.16	….	….	….	0.15	….	….	….	0.16
ρ	….	….	….	….	….	….	….	….	0.45	….	….	….	0.14	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.42	….	….	….	0.15	….	….	….	0.83
AIC	….	….	….	….	123.91	….	….	….	124.05	….	….	….	126.10	….	….	….	125.13	….	….	….	125.77	….	….	….	124.55

Table 7:

Coefficient estimates and SE for the OLS, SLM, SDM, SEM, CAR, and SARMA multiple regression models and associated VIF and P values for disease rating regressed against Phytophthora rubi DNA in roots, Pratylenchus penetrans in roots and soil in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables^a	OLS					SLM				SDM				SEM				CAR				SARMA
Field	Variables^a	Estimate	SE	P	VIF	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model
OR-1	Intercept	0.83	0.03	<0.001	….	….	0.20	0.14	0.15	….	0.78	0.34	0.02	….	0.83	0.05	<0.001	….	0.82	0.03	<0.001	….	0.83	0.05	<0.001	….
	P. rubi DNA	0.08	0.01	<0.001	1.02	….	0.07	0.01	<0.001	….	0.07	0.01	<0.001	….	0.07	0.01	<0.001	….	0.09	0.01	<0.001	….	0.07	0.01	<0.001	….
	P. penetrans (root)	0.01	0.02	0.57	1.30	….	0.01	0.02	0.52	….	0.02	0.02	0.20	….	0.01	0.02	0.62	….	0.01	0.02	0.71	….	0.01	0.02	0.66	….
	P. penetrans (soil)	−0.04	0.02	0.04	1.32	….	−0.02	0.02	0.15	….	−0.03	0.02	0.10	….	−0.02	0.02	0.19	….	−0.03	0.02	0.04	….	−0.01	0.01	0.42	….
	P	….	….	….	….	<0.001	….	….	….	<0.001	….	….	….	0.79	….	….	….	<0.001	….	….	….	0.56	….	….	….	<0.001
R²	….	….	….	….	0.46	….	….	….	0.57	….	….	….	0.61	….	….	….	0.53	….	….	….	0.46	….	….	….	0.56
ρ	….	….	….	….	….	….	….	….	0.67	….	….	….	−0.12	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.73	….	….	….	0.09	….	….	….	2.06
AIC	….	….	….	….	−84.58	….	….	….	−95.82	….	….	….	−96.37	….	….	….	−90.26	….	….	….	−82.92	….	….	….	−95.25
OR-2	Intercept	0.61	0.02	<0.001	….	….	0.58	0.23	0.01	….	0.74	0.25	0.003	….	0.61	0.02	<0.001	….	0.61	0.02	<0.001	….	0.61	0.02	<0.001	….
	P. rubi DNA	0.01	0.01	0.27	1.01	….	0.01	0.01	0.25	….	0.006	0.01	0.56	….	0.01	0.01	0.25	….	0.01	0.01	0.26	….	0.01	0.01	0.25	….
	P. penetrans (root)	0.008	0.006	0.22	1.09	….	0.008	0.006	0.21	….	0.01	0.006	0.08	….	0.008	0.006	0.18	….	0.008	0.006	0.19	….	0.008	0.006	0.18	….
	P. penetrans (soil)	0.01	0.01	0.38	1.09	….	0.01	0.01	0.36	….	0.004	0.01	0.74	….	0.01	0.01	0.35	….	0.01	0.01	0.35	….	0.01	0.01	0.34	….
	P	….	….	….	….	0.21	….	….	….	0.88	….	….	….	0.58	….	….	….	0.81	….	….	….	0.95	….	….	….	0.77
R²	….	….	….	….	0.08	….	….	….	0.08	….	….	….	0.17	….	….	….	0.08	….	….	….	0.08	….	….	….	0.08
ρ	….	….	….	….	….	….	….	….	0.05	….	….	….	−0.24	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.10	….	….	….	−0.01	….	….	….	−0.13
AIC	….	….	….	….	−153.46	….	….	….	−151.48	….	….	….	−152.09	….	….	….	−151.52	….	….	….	−151.47	….	….	….	−151.54
WA-1	Intercept	0.59	0.03	<0.001	….	….	0.31	0.13	0.01	….	0.82	0.27	0.003	….	0.58	0.04	<0.001	….	0.59	0.03	<0.001	….	0.58	0.04	<0.001	….
	P. rubi DNA	0.09	0.01	<0.001	1.00	….	0.08	0.01	<0.001	….	0.08	0.01	<0.001	….	0.08	0.01	<0.001	….	0.09	0.01	<0.001	….	0.08	0.01	<0.001	….
	P. penetrans (root)	−0.01	0.03	0.63	1.01	….	−0.008	0.02	0.74	….	−0.02	0.02	0.48	….	−0.009	0.02	0.74	….	−0.01	0.03	0.59	….	−0.008	0.02	0.73	….
	P. penetrans (soil)	−0.05	0.02	0.03	1.01	….	−0.03	0.02	0.18	….	−0.02	0.03	0.51	….	−0.04	0.02	0.11	….	−0.05	0.02	0.04	….	−0.038	0.03	0.14	….
	P	….	….	….	….	<0.001	….	….	….	0.04	….	….	….	0.36	….	….	….	0.35	….	….	….	0.83	….	….	….	0.28
R²	….	….	….	….	0.43	….	….	….	0.46	….	….	….	0.51	….	….	….	0.43	….	….	….	0.43	….	….	….	0.44
ρ	….	….	….	….	….	….	….	….	0.44	….	….	….	−0.48	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.38	….	….	….	0.04	….	….	….	0.63
AIC	….	….	….	….	−54.35	….	….	….	−56.27	….	….	….	−55.19	….	….	….	−53.22	….	….	….	−52.40	….	….	….	−53.52
WA-2	Intercept	0.53	0.05	<0.001	….	….	0.09	0.09	0.27	….	0.47	0.18	0.009	….	0.59	0.09	<0.001	….	0.57	0.05	<0.001	….	0.62	0.06	<0.001	….
	P. rubi DNA	0.12	0.01	<0.001	1.07	….	0.08	0.01	<0.001	….	0.08	0.01	<0.001	….	0.08	0.01	<0.001	….	0.12	0.01	<0.001	….	0.08	0.01	<0.001	….
	P. penetrans (root)	0.01	0.01	0.44	1.14	….	−0.009	0.01	0.39	….	−0.01	0.01	0.22	….	−0.009	0.01	0.40	….	0.01	0.01	0.44	….	−0.01	0.01	0.20	….
	P. penetrans (soil)	0.07	0.02	0.006	1.08	….	0.04	0.02	0.02	….	0.05	0.02	0.009	….	0.05	0.02	0.01	….	0.04	0.02	0.08	….	0.07	0.02	<0.001	….
	P	….	….	….	….	<0.001	….	….	….	<0.001	….	….	….	0.60	….	….	….	<0.001	….	….	….	0.18	….	….	….	<0.001
R²	….	….	….	….	0.57	….	….	….	0.70	….	….	….	0.73	….	….	….	0.68	….	….	….	0.58	….	….	….	0.68
ρ	….	….	….	….	….	….	….	….	0.67	….	….	….	0.20	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	0.84	….	….	….	0.31	….	….	….	2.06
AIC	….	….	….	….	−76.74	….	….	….	−97.91	….	….	….	−98.44	….	….	….	−92.92	….	….	….	−76.50	….	….	….	−93.94

a Independent variables were P. rubi DNA concentrations in roots and P. penetrans in roots and soil. The P value represents the significance of the model at P < 0.05. R² is the Nagelkerke pseudo-R-squared; ρ is a fitting parameter for SLM and SDM; λ is a fitting parameter for SEM, CAR, and SARMA. AIC was used to compare OLS, SLM, SDM, SEM, CAR, and SARMA models and determine the best model that fits the data for each field. The lower the AIC, the better the model.

Table 8:

Coefficient estimates and SE for the OLS, SLM, SDM, SEM, CAR, and SARMA multiple regression models and associated VIF and P values for Pratylenchus penetrans densities in roots regressed against P. penetrans densities in soil and Phytophthora rubi DNA concentrations in roots in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Variables^a	OLS					SLM				SDM				SEM				CAR				SARMA
Field	Variables^a	Estimate	SE	P	VIF	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model	Estimate	SE	P	Model
OR-1	Intercept	0.11	0.19	0.59	….	….	0.05	0.26	0.84	….	−1.19	1.28	0.35	….	0.10	0.19	0.60	….	0.11	0.19	0.57	….	0.09	0.19	0.62	….
	P. penetrans (soil)	0.39	0.09	<0.001	1.02	….	0.38	0.09	<0.001	….	0.42	0.10	<0.001	….	0.39	0.09	<0.001	….	0.39	0.09	<0.001	….	0.40	0.09	<0.001	….
	P. rubi DNA	0.02	0.08	0.82	1.02	….	0.02	0.07	0.79	….	0.06	0.08	0.46	….	0.02	0.07	0.83	….	0.01	0.07	0.87	….	0.02	0.07	0.83	….
	P	….	….	….	….	<0.001	….	….	….	0.75	….	….	….	0.56	….	….	….	0.83	….	….	….	0.85	….	….	….	0.75
R²	….	….	….	….	0.23	….	….	….	0.23	….	….	….	0.25	….	….	….	0.23	….	….	….	0.23	….	….	….	0.23
ρ	….	….	….	….	….	….	….	….	0.11	….	….	….	−0.27	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.10	….	….	….	−0.14	….	….	….	−0.18
AIC	….	….	….	….	126.36	….	….	….	128.26	….	….	….	130.39	….	….	….	128.32	….	….	….	128.33	….	….	….	128.26
OR-2	Intercept	0.32	0.48	0.52	….	….	0.55	0.70	0.43	….	−3.38	1.69	0.04	….	0.19	0.44	0.66	….	0.07	0.44	0.86	….	−0.09	0.25	0.72	….
	P. penetrans (soil)	0.59	0.7	0.03	1.0	….	0.60	0.26	0.02	….	0.64	0.25	0.01	….	0.64	0.25	0.01	….	0.69	0.25	0.005	….	0.71	0.11	<0.001	….
	P. rubi DNA	0.10	0.22	0.67	1.0	….	0.10	0.22	0.63	….	0.26	0.21	0.22	….	0.14	0.22	0.49	….	0.15	0.22	0.49	….	0.39	0.21	0.06	….
	P	….	….	….	….	0.08	….	….	….	0.61	….	….	….	0.09	….	….	….	0.36	….	….	….	0.35	….	….	….	<0.001
R²	….	….	….	….	0.08	….	….	….	0.09	….	….	….	0.17	….	….	….	0.09	….	….	….	0.09	….	….	….	0.33
ρ	….	….	….	….	….	….	….	….	−0.21	….	….	….	−0.85	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.44	….	….	….	−1.02	….	….	….	−0.99
AIC	….	….	….	….	206.40	….	….	….	208.14	….	….	….	206.19	….	….	….	207.56	….	….	….	207.53	….	….	….	189.57
WA-1	Intercept	0.48	0.14	0.001	….	….	0.51	0.25	0.04	….	−0.03	0.43	0.94	….	0.46	0.13	<0.001	….	0.48	0.13	<0.001	….	0.45	0.12	<0.001	….
	P. penetrans (soil)	0.09	0.12	0.42	1.0	….	0.10	0.11	0.38	….	−0.04	0.15	0.81	….	0.11	0.11	0.30	….	0.09	0.11	0.42	….	0.12	0.10	0.22	….
	P. rubi DNA	−0.01	0.07	0.85	1.0	….	−0.01	0.07	0.85	….	−0.006	0.07	0.93	….	−0.01	0.07	0.87	….	−0.01	0.07	0.88	….	−0.01	0.06	0.87	….
	P	….	….	….	….	0.71	….	….	….	0.88	….	….	….	0.69	….	….	….	0.72	….	….	….	0.78	….	….	….	0.65
R²	….	….	….	….	0.01	….	….	….	0.01	….	….	….	0.06	….	….	….	0.01	….	….	….	0.01	….	….	….	0.01
ρ	….	….	….	….	….	….	….	….	−0.06	….	….	….	−0.17	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.17	….	….	….	−0.21	….	….	….	−0.27
AIC	….	….	….	….	134.05	….	….	….	136.03	….	….	….	137.17	….	….	….	135.92	….	….	….	135.97	….	….	….	135.84
WA-2	Intercept	0.80	0.49	0.11	….	….	0.61	0.73	0.40	….	1.06	1.35	0.43	….	0.59	0.43	0.18	….	0.69	0.47	0.14	….	0.70	0.46	0.13	….
	P. penetrans (soil)	0.47	0.22	0.04	1.0	….	0.45	0.22	0.04	….	0.34	0.21	0.10	….	0.52	0.20	0.01	….	0.53	0.21	0.01	….	0.50	0.21	0.02	….
	P. rubi DNA	0.26	0.14	0.06	1.0	….	0.24	0.14	0.08	….	0.08	0.14	0.58	….	0.36	0.12	0.003	….	0.26	0.13	0.05	….	0.31	0.13	0.02	….
	P	….	….	….	….	0.03	….	….	….	0.66	….	….	….	0.06	….	….	….	0.51	….	….	….	0.71	….	….	….	0.66
R²	….	….	….	….	0.12	….	….	….	0.12	….	….	….	0.29	….	….	….	0.13	….	….	….	0.12	….	….	….	0.12
ρ	….	….	….	….	….	….	….	….	0.13	….	….	….	−0.82	….	….	….	….	….	….	….	….	….	….	….	….
λ	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	….	−0.40	….	….	….	−0.24	….	….	….	−0.16
AIC	….	….	….	….	190.27	….	….	….	192.08	….	….	….	183.53	….	….	….	191.85	….	….	….	192.14	….	….	….	192.08

a Independent variables were P. rubi DNA concentrations in roots and P. penetrans in soil. The P value represents the significance of the model at P < 0.05. R² is the Nagelkerke pseudo-R-squared; ρ is a fitting parameter for SLM and SDM; λ is a fitting parameter for SEM, CAR, and SARMA. AIC was used to compare OLS, SLM, SDM, SEM, CAR, and SARMA models and determine the best model that fits the data for each field. The lower the AIC, the better the model.

Table S1:

Global Moran’s I test for spatial autocorrelation in the residuals for Phytophthora rubi DNA concentrations in roots, Pratylenchus penetrans densites in root and soil regressed against soil texture (sand, silt, and clay), and field elevation in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Predicted	Moran’s I coefficient^a	P-value^b
OR-1	P. rubi DNA in root	−0.018	0.29
	P. penetrans in root	−0.026	0.39
	P. penetrans in soil	−0.030	0.45
OR-2	P. rubi DNA in root	−0.011	0.39
	P. penetrans in root	−0.016	0.44
	P. penetrans in soil	0.046	0.03
WA-1	P. rubi DNA in root	−0.017	0.36
	P. penetrans in root	−0.001	0.19
	P. penetrans in soil	0.100	<0.001
WA-2	P. rubi DNA in root	0.064	<0.001
	P. penetrans in root	−0.060	0.78
	P. penetrans in soil	0.030	0.01

a Global Moran’s I coefficient values range from −1 to 1, where a value <0 indicates negative spatial autocorrelation and a value >0 indicates positive spatial autocorrelation.

b The P-values indicate the level of significance at P < 0.05 for each coefficient.

Table S2:

Global Moran’s I test for spatial autocorrelation in the residuals for visual disease rating regressed against Phytophthora rubi DNA concentrations in roots and Pratylenchus penetrans densities in roots and soil in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Moran’s I coefficient^a	P-value^b
OR-1	0.13	<0.001
OR-2	−0.01	0.41
WA-1	0.03	0.05
WA-2	0.19	<0.001

a Global Moran’s I coefficient values range from −1 to 1, where a value <0 indicates negative spatial autocorrelation and a value >0 indicates positive spatial autocorrelation.

b The P-values indicate the level of significance P < 0.05 for each coefficient.

Table S3:

Global Moran’s I test for spatial autocorrelation in the residuals for Pratylenchus penetrans densities in root regressed against P. penetrans densities in soil and Phytophthora rubi DNA concentrations in roots in four commercial red raspberry production fields in Oregon (OR-1 and OR-2) and Washington (WA-1 and WA-2).

Field	Moran’s I coefficient^a	P-value^b
OR-1	−0.01	0.39
OR-2	−0.03	0.66
WA-1	−0.01	0.36
WA-2	−0.02	0.45

a Global Moran’s I coefficient values range from −1 to 1, where a value <0 indicates negative spatial autocorrelation and a value >0 indicates positive spatial autocorrelation.

b The P-values indicate the level of significance P < 0.05 for each coefficient.

References

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Articles in this issue

DOI: https://doi.org/10.2478/jofnem-2025-0038 | Journal eISSN: 2640-396X | Journal ISSN: 0022-300X

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Language: English

Submitted on: Jan 2, 2025

Published on: Sep 24, 2025

Published by: Society of Nematologists, Inc.

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

ecology and epidemiology,

Phytophthora root rot,

quantitative epidemiology,

root lesion nematode nematode,

spatial autocorrelation,

spatial autoregressive model

Related subjects:

Life sciences,

Life sciences, other

© 2025 J. B. Contina, D. R. Kroese, T. W. Walters, J. E. Weiland, I. A. Zasada, published by Society of Nematologists, Inc.
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 57 (2025): Issue 1 (February 2025)

Multivariate Geostatistical Modeling of Phytophthora rubi and Pratylenchus penetrans in Red Raspberry Fields

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