In recent years, statistical methods involving spatial considerations have been developed, for example, those incorporating data with some type of georeferencing. The descriptive part of geographic information systems currently provides many visualization and analytic tools; however, the latter is still quite limited. In this sense, research of a spatial nature is seen as combining non-spatial statistical methods for inferential treatment that can certainly invalidate the excellent capture work with advanced tools such as those observed every day in the geomatic context. This prompted the current document, drawing attention to how geomatic information analyzed with statistical methods that imply independence in modeled observations can be invalid. The Moran index is compared with a proposal for a spatial lag coefficient in the context of experimental design so that users of variance analysis do not apply this well-known procedure in a ritualistic way, perhaps revising some assumptions and perhaps ignoring more important ones. The distortion of the p value generated from the analysis of variance is clear in the presence of spatial dependence. In this case it is associated with the lag or spatial overlap. The methodology is simple to adopt in other experimental designs with the simple consideration of the design matrix and its reparameterization and the choice of the appropriate weight matrix. This will allow users to reconsider the traditional method of analysis and incorporate some methodology to support spatial dependency structures.
Research Article
The effect of spatial lag on modeling geomatic covariates using analysis of variance
https://doi.org/10.21203/rs.3.rs-3243407/v1
This work is licensed under a CC BY 4.0 License
Journal Publication
published 22 Jul, 2024
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spatial lag
spatial overlapping
spectral index
sensing
Analysis of variance (ANOVA) is a very useful tool in the biological and industrial context, and it has undoubtedly gained ground in modern fields such as geomatics, where data captured with remote or proximal sensors predominate. ANOVA has been one of the standard tools used to analyze experimental data for decades; however, currently there are so many application modalities that it is not enough to appear in the methodology that applied an ANOVA (AOV, ANAVAR, ANDEVA) to have complete clarity of what is behind the method and its proper use, since aspects such as balancing, blocking, the presence of covariates, temporal measurements, geo-referenced information, discrete or categorical response(s) of univariate or multivariate nature generate their own analytic strategies for estimating parameters of the associated model (Christensen, 2019).
Currently, remote sensing data capture technologies, and drone images or special cameras allow the creation of spectral indexes (Adak et al., 2021) that are associated with data that usually have very different structure to the experimental design, obligating users to do some kind of aggregation to experimental plots or other units in order to be able to treat this information as a source of additional variation in the ANOVA or simply as the same response. These data that involve some form of georeferencing, may also incorporate some spatial dependency structure (and many other temporal and space-times). Regarding different aspects such as analysis of vegetation cover, climatic and meteorological variables, among others, it is important to highlight that these vary due to many factors, including the time scale, therefore, it is necessary to develop evaluations with updated information from two or more periods, which allow comparisons to identify temporary variations of the object of study, before the attribution of the cause of said change. This process is known as multitemporal evaluation, which consists of comparing the object in the same geographical area in different periods, in order to know its variation, for this there are several evaluation methods (Gao et al., 2012). If these aspects are overlooked in the inferential process, it could invalidate its application, especially if they are analyzed with the methods usually studied in basic courses of experimental design (Kenny, 1995; Kenny & Judd, 1986; Zimmerman et al., 2007; Zimmerman & Zumbo, 2010).
It is surprising that many users of the technique incompletely report aspects for consideration prior to the ANOVA (experimental design) in the ANOVA’s own results; and they report insufficient details of analysis after ANOVA (interactions or multiple comparisons) to be certain of the scope and interpretation of the results (Acutis et al., 2012). This allows its statistical software package to set the tone for substantive data analysis decision-making usurping in some way the preponderant role of the users, perhaps assuming that by the fame of the software or its recognition it can be taken for granted that everything developed there is directly applicable or interpretable.
Several documents highlight the misuse of ANOVA in two-way designs due to heterogeneous variance (Rowell & Walters, 2006) highlighting defective data (Pearce, 2006). Defective data includes the non-specification of the type of ANOVA (Abubakar et al., 2022), confusion in the choice of the type of effects (fixed or random) (Bennington & Thayne, 1994), assumption too quickly of the weak argument that not rejecting the null hypothesis implies a strong conclusion (Rong, 2000), inappropriate use of the terms associated with the error in the construction of the statistical F or the type of sums of squares (Li & Lomax, 2011). Such defective data even includes the belief, supported in recognized literature that only a couple of assumptions are necessary to make use of the technique (Acutis et al., 2012). This ignores the supposed key to independence (Lindman, 1992) in which it is closely linked to the reality of the data captured with sophisticated sensors or equipment related to agriculture, such as performance monitors, spectrometric or other techniques for capturing data from radar or satellite images, or images from spectral or hyperspectral cameras, etc. All this information is usually converted into indices that usually relate to the physical-chemical and microbiological properties of soils or water as well as the vegetation, specifically in the development of the crop. In turn, this information is quite often incorporated into the structure of the experimental design in order to associate the data with each observation or experimental unit and thus, using ANOVA, evaluate the effect of the treatments (Atik & Akdemir, 2022; Barbosa et al., 2020; Ding et al., 2020; Firozjaei et al., 2020; Stoy et al., 2022; Volcani et al., 2005; Ya’acob et al., 2014). In the documents cited above, how SIMPLE (Simple, Informative, Meaningful, Powerful, Logical, Effective,) (Mcintosh, 2015) were these analyses in general? In the case of data which in many cases have been shown to have spatial or temporal dependency, shouldn´t an adequate description be made of the essential aspects of the ANOVA to involve the dependency structure present in the data? Knowing the weaknesses of ANOVA in certain contexts and of certain types of data can very surely enhance its benefits.
A form of spatial dependence can arise with remote sensor data or similar technologies, such as spatial lag or overlap (Shukl & Subrahmanyam, 1999), in order to find data associated with experiments where some design model was adjusted for the comparison of treatments, using as response(s) or some performance indicator evaluated in field or some estimate generated by some type of sensor (Wang et al., 2016). Some of these indices are added to each polygon that make up the experimental plot to finally incorporate them into the model, sometimes as a response and sometimes as covariates (if treatments do not affect this covariate, or a multiple-slope model has been specified). The literature points out how these indices show some kind of spatial dependence (dos Santos et al., 2021; H. Zhang et al., 2011); however, the rituality of traditional analysis mainly by knowledge of the methods that usually do not treat spatial dependence allows many excellent data captures, invalidating the work of field or laboratory tests because of the analytic technique (Gotway & Cressie, 1990).
With all the reasons and documentation presented, it is important to understand how ANOVA can be invalidated, at least in the context of spatial lag; but the same results extend to other situations that may have the same effect, such as edge effects, competition, poor specification of a model, etc. (Christensen & Bedrick,1997). Using conditional simulations, a Gaussian variogram was generated with fully defined parameters, and 1000 simulations were obtained to generate spatial dependence. The final goal was to show the invalidity of the ANOVA when spatial lag was presented using the Moran index and the spatial lag coefficient, although the other assumptions usually found as the most important in the context of the ANOVA are fulfilled.
In the linear models treated in experiment design, Gauss-Markov dominates, written as \(\mathbf{y} = \mathbf{X}\varvec{\beta } + \varvec{\epsilon }\) ; \(E\left(\varvec{\epsilon }\right)=0\) and \(Cov\left(\varvec{\epsilon }\right)={}^{2}\varvec{V}\), where \(\varvec{V}\) is a known matrix representing some correlation structure between residuals. The different models depend on the assumptions made about \(\varvec{V}\), and it is precisely the simplest one dealt with in the design where it is assumed that V = I (an identity matrix), that is, it assumes independence and identical distribution, something that cannot be assumed so quickly when data is associated with information from remote sensing technologies. To deal with this limitation of the model, an extended model by spatial lag is used written as:
$$\mathbf{y} = \mathbf{X}\varvec{\beta } + {{\alpha }}_{\text{s}}\mathbf{W} \mathbf{X}\varvec{\beta } + \varvec{\epsilon }$$
1
where \(\mathbf{y}\) is a random vector of length n denoting the response of n experimental units where the response associated with the spectral index, or any other index could usually be involved. \(\mathbf{X}\) is the known design matrix of dimension n×p and incomplete range (as a design model), \(\varvec{\beta }\) is a vector of unknown parameters of length p, which, in the case that develops consists of the effect of blocks and treatments, \({{\alpha }}_{\text{s}}\)is the coefficient associated with the effect of spatial lag (overlap) and \(\mathbf{W}\) = (wij ) is the matrix of spatial weights of dimension n×n, where wij represents the effect of unit j on unit i. A conditional simulation was generated from a Gaussian variogram (Gräler et al, 2019) with parameters as described in the following R script:
The 1000 simulations were assigned to a 10m×6m grid for 60 polygon-associated cells representing the experimental units. With this matrix and their respective latitude and longitude coordinates artificially created for the grid (per row[y] and column[x]), three treatments were assigned in two blocks to associate the ANOVA to the simple factorial design in generalized and random complete blocks without interaction. The random generation of this data set ensured that in the 1000 runs of the ANOVA there was no effect of the treatments nor that the blocking was efficient in not distorting the distribution of the test statistic (F) as regards the no centrality parameter. The image generated for the grid with treatment-block-repeat assignment (T.B.R) is shown in Fig. 1.
Each cell associated with each polygon was labeled with the respective treatment (T:{1,2,3}), blocking ratio (B:{1,2}), and repetition (R:{1,2,...,10}). To generate the spatial lag, the cells were spatially varied (exchanging) by two in two in the combinations of T.B.R within the grid to modify the distance between these units and change the pattern of the neighborhood and, therefore, of the spatial dependence.
In Fig. 2, each image with different ordering (ord0[data from the original variogram with high spatial dependence] up to ord60 [data with all grid elements exchanged and thus with low spatial dependence]) represents the same T.B.R response.
For all systems, the Moran index was calculated (J. Zhang et al., 2014) and applied to the residual design model once the ANOVA ran in all simulations, thus creating different scenarios of spatial dependence. This index was calculated with the ape library of R (Paradis & Schliep, 2019).
In each simulation, the ANOVA F statistic was extracted, normality and homoscedasticity of residues were tested for treatments. The same OT lag statistic was calculated based on the Chi-square distribution (with a degree of freedom) that appears in Darghan et al. (2012) and that is annexed in this document in a script of R for use, as well as the coefficient associated with the spatial lag (\({{\alpha }}_{\text{s}}\)) which was obtained from:
$${ \widehat{{\alpha }}}_{\text{s}}=\frac{\mathbf{y}\mathbf{{\prime }}{\mathbf{M}}_{0}\varvec{W}(\mathbf{I}-{\mathbf{M}}_{0})\mathbf{y}}{\varvec{y}\varvec{{\prime }}{\mathbf{M}}_{0}\varvec{W}\varvec{W}{\mathbf{M}}_{0}\varvec{y}} \left(2\right)$$
where \({\mathbf{M}}_{0}\) is associated with the design matrix having already repaired the model in the Christensen (2011) style, with \({\mathbf{M}}_{0}={\varvec{X}}_{0}{\left({\varvec{X}}_{0}\varvec{{\prime }}{\varvec{X}}_{0}\right)}^{-1}{\varvec{X}}_{0}^{\varvec{{\prime }}}\) where \({\varvec{X}}_{0}=\varvec{X}\varvec{Z}\) being \(\varvec{Z}\) generated with the "mz" function of R available in (GitHub1: supplementary material). This matrix depends on the selected design, and without loss of generality, it will be enough to modify the design matrix (in balanced cases) and thus obtain this special matrix for the new design. The equation in (2) also depends on the weight matrix and in this case the inverses of the Euclidean distances were used with type C standardization according to R spdep library (Bivand, 2022). For all spatial variations all these measurements were calculated in addition to the ANOVA (AOV function of R of the library stats (R Core Team, 2022)), where the residuals of the model were extracted to test normality (function Shapiro.test of the library stats of R) and homoscedasticity test in the treatments (Bartlett.test function of R stats). All the information formed a dimension matrix (31000×7) that appears available in github2 (supplementary material- 31 ordering scenarios for 1000 simulations and seven responses). With all these data diagrams that were made that related the p value of the Moran index and p value of the ANOVA statistic F with the statistic OT, the spatial lag coefficient in each treatment was arranged to demonstrate how this order was related to the Moran index and the lag coefficient, thus highlighting the little or lack of usefulness of the ANOVA in the presence of a dependence or spatial lag.
Figure 3 is one of the most important in this document because it shows the effect of the spatial lag in the distributional behavior of the F statistic. The monotonous continuous line represents the theoretical distribution of the F statistic (central) with 2 and 56 degrees of freedom (for the established design). The dotted vertical line represents the quantile value for a 95% confidence level and the x-axis represents the values of the F obtained in the 1000 simulations. The histogram describes the observed distributional behavior of the F statistic in the presence of spatial lag. It is evident how the theoretical curve F cannot be used to obtain the p value of the ANOVA even though practically all the generated F values do not reject the null hypothesis of the null effect of treatments. Even in the presence of spatial lag, both scenarios lead to no rejection, but the p value, usually generated from the ANOVA is not the measure that represents what is occurring. This would also apply if the rejection of the null hypothesis was even due to a difference in treatment means.
Figure 4 shows the probabilities associated with theoretical F and simulated F in the ord0, ord30 and ord60 systems to indicate the spatial dependence according to the Moran index. Already in the last order the spatial lag has been lost and therefore the p value of Moran is greater than 5% in almost all simulations. The dotted line represents the points where both probabilities coincide and happens when the statistic F in both cases tends to zero. It is also evident that it is more likely not to reject the null hypothesis of equality of means in spatial lag scenarios, making it easier to obtain false negatives in the presence of this effect.
Figure 5 is like Fig. 4 except that the coefficient associated with the spatial lag for the same orders (ord0, ord30 and ord60) is highlighted here. The estimated coefficient of lag is near zero in spatial variations of the greatest occurrence of re-orders.
Figure 6 shows the OT statistic for spatial lag for the null hypothesis Hoαs = 0. This hypothesis is tested with one degree of freedom. Also shown is a vertical line (for the 95% confidence level) in different systems to delimit in the x-axis the quantum of the theoretical F. The crossing of these lines yields four regions: A) [spatial lag and no treatment effect], B) [spatial lag and no treatment effect], C) [spatial lag and no treatment effect] and D) [spatial lag and no treatment effect]. From orders higher than or equal to 36 (ord36) practically all the data fall in region B of absence of spatial lag and null effect of the treatments. In the ord0, where the data with Moran index indicate spatial dependence, for a fraction of these data there is a spatial lag (B) and in the other there is none (A). To better understand these results, Table 1 describes the proportion of data in each region.
| Region | ||||
|---|---|---|---|---|
| Order | A | B | C | D |
| Ord0 | 49.7 | 50.2 | 0.0 | 0.1 |
| Ord12 | 37.9 | 62 | 0.0 | 0.1 |
| Ord24 | 30.1 | 69.8 | 0.1 | 0.0 |
| Ord36 | 2.7 | 97.2 | 0.0 | 0.1 |
| Ord48 | 2.8 | 97.1 | 0.0 | 0.1 |
| Ord60 | 1.3 | 98.6 | 0.0 | 0.1 |
From Table 1 it is evident that from the Ord36 most of the data falls into B and at most 10% between regions C and D. This clearly shows as the spatial lag is lost; the spatial dependence is lost even though the ANOVA shows no evidence in the data to reject the null hypothesis of null effect.
Finally, Fig. 7 separates by ordering the behavior of the estimated spatial lag coefficient \({\widehat{\alpha }}_{s}\) where an average at zero and greater variability is clearly perceived when there is greater spatial dependence and thus spatial lag. The clearest points indicate the cases with no spatial dependence when the estimated values of \({\widehat{\alpha }}_{s}\)oscillate approximately between − 0.01 and 0.01, maintaining a similar variability since the ord36, when the Moran index already suggests spatial independence.
Although ANOVA yields in practice results associated with not rejecting the hypothesis of the null effect of treatments, the p value of the test is not appropriate as it does not reflect the probability of obtaining test results for the F statistic, at least as extreme as the result observed, on the assumption that the null hypothesis is correct.
Spatial dependence has a direct impact on ANOVA validity. This violation of the assumption of independence by spatial lag (although it can be violated in many ways (Christensen, 1997) may have several important implications in the ANOVA, highlighting here the underestimation of the p-value of Exhibit F that affects the validity of the associated hypothesis tests and thus the interpretation of results, since the spatial dependence introduces a structure in the data that is not taken into account in the conventional ANOVA that is usually taught in basic courses of experimental design.
Although there are different methods for detecting spatial dependency, surely the Moran index or some of its modifications are the most used and are already programmed in several statistical packages. In the case of the lag coefficient, it could also be a convenient measure adapted to each experimental design using the design matrix. As this represents the ratio of two independent quadratic forms, it is possible not only to obtain their moments but even their distributional behavior, making this a measure of alternative interest to judge spatial lag in the context of experimental design, provided that a weight matrix is defined ensuring symmetry in the matrices associated with the quadratic forms.
Spatial dependence represents a significant violation of the assumption of independence of the residues in the ANOVA. This assumption assumes that observations are independent of each other, that is, that the value of an observation is not related to the value of another nearby observation in space. When spatial dependence exists, close observations tend to be more like each other than more distant observations, meaning that the residues of neighboring observations are correlated and do not meet the assumption of independence. Since the observations taken by remote or proximal sensors of physic-chemical or microbiological properties of soils and vegetation can be grouped in space by evolutionary processes, degradation, crop or soil management or other agents can generate similar predictor variables. This similarity generates a structure in the correlation of observations that manifests as spatial dependence. If this dependence is not considered and in this case is caused by spatial lag in treatments, there is a risk of obtaining incorrect estimates of the p value associated with the ANOVA F test statistic; therefore, this leads to inappropriate inferences and, therefore, to interpretations of little or no value. Spatial autocorrelation can lead to overestimation or underestimation of the significance of treatment effects that can lead to erroneous acceptance or incorrect rejection of associated hypotheses.
Addressing spatial dependence in the analysis of variance by Moran's index or by the lag coefficient for a specified experimental design can help to identify the problem and to take the necessary measures using the appropriate procedures or models for an estimate of some response or an appropriate comparison of treatments.
Spatial dependence represents a critical violation of the independence assumption in the ANOVA and should not be omitted in analyses where studies have proven some form of dependence, as is the case in many of the phenomena that are modeled in the context of geomatics.
Conflict of Interest
The authors declare that there is no conflict of interest.
Data Availability
The data as well as the R scripts to generate the simulations can be found at:
GitHub 1: https://github.com/CarlosRivera1212/rezago_aov
GitHub 2: https://github.com/darklley/Rezago
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