A discussion on the significance associated with Pearson s correlation in precision agriculture studies

Transcription

1 Precision Agric (2013) 14: DOI /s SHORT DISCUSSSION A discussion on the significance associated with Pearson s correlation in precision agriculture studies J. A. Taylor T. R. Bates Published online: 23 April 2013 Ó Springer Science+Business Media New York 2013 Abstract Pearson s correlation is a commonly used descriptive statistic in many published precision agriculture studies, not only in the Precision Agriculture Journal, but also in other journals that publish in this domain. Very few of these articles take into consideration auto-correlation in data when performing correlation analysis, despite a statistical solution being available. A brief discussion on the need to consider auto-correlation and the effective sample size when using Pearson s correlation in precision agriculture research is presented. The discussion is supported by an example using spatial data on vine size and canopy vigour in a juice-grape vineyard. The example data demonstrated that the p-value of the correlation between vine size and canopy vigour increased when auto-correlation was accounted for, potentially to a non-significant level depending on the desired a-level. The example data also demonstrated that the method by which data are processed (interpolated) to achieve co-located data will also affect the amount of auto-correlation and the effective sample size. The results showed that for the same variables, with different approaches to data co-location, a lower r-value may have a lower p-value and potentially hold more statistical significance. Keywords Auto-correlation Pruning weight NDVI Introduction The availability of spatial data-sets within the agricultural and environmental sciences has increased dramatically over the past two decades. However, spatial data do not hold true to some of the assumptions in conventional, non-spatial analyses that agricultural scientists are most familiar with. A key assumption for many data analyses, including Pearson s correlation, is the assumption of the independence of the data and that the errors are independent and identically distributed (i.i.d.). However, there is usually some level of J. A. Taylor (&) T. R. Bates Cornell Lake Erie Research and Extension Laboratory, Department of Horticulture, Cornell University, 6592 West Main St, Portland, NY 14769, USA james.taylor@cornell.edu

2 Precision Agric (2013) 14: auto-correlation in spatial data that reduces the independence of the data, causing the errors to be incorrectly modelled and to violate the i.d.d. assumption. While this is true for many non-spatial analyses, this article deals specifically with the issue of auto-correlation in data in relation to Pearson s correlation, which is an often used descriptive statistic in the Precision Agriculture (PA) literature. The confidence interval and p-value in Pearson s correlation is dependent on the sample size (n), which is used in the calculation of the standard error component of the relevant equations. The sample size is considered to be the number of independent observations within a data-set. For variables that exhibit spatial auto-correlation, observations within a certain neighbourhood will be correlated, i.e. not independent of each other. Therefore, the assumption that each observation is independent is violated and an adjustment must be made to determine the actual number of independent observations, which is considered the effective sample size (n eff ), before performing Pearson s correlation. Furthermore, data collected in PA studies are often derived from different platforms and/or at different spatial resolutions and requires some form of interpolation to co-locate different data-sets. Correct interpolation procedures will increase the amount of auto-correlation in the data and, depending on the grid size chosen, may also increase the n (the number of grid points) used for the correlation analysis above the sampling density (number of observations in the raw data). Consequently, an uncorrected correlation analysis of spatially structured data, particularly interpolated data that relies on n rather than n eff is statistically flawed. However, for co-located spatial data, methods do exist to determine n eff that take into account the spatial structure of both variables (Clifford et al. 1989; Dutilleul 1993). These methods utilise the sample covariance to adjust the sample size to take account of any autocorrelation in the spatial processes. The approach of Clifford et al. (1989) is an approximation, while Dutilleul (1993) presents a formal mathematical derivative. Dutilleul s approach is computationally more intensive but more accurate, especially with small data networks (Dutilleul 1993). Current computing power means that Dutilleul s method is feasible for most data-sets. These corrections for n eff appear to be used in ecological studies but not in precision agricultural studies. A search of the SciVerse Scopus database found that the work of Clifford et al. (1989) and Dutilleul (1993) has never been cited in the Precision Agriculture Journal, although Pearson s correlation with an associated significance is commonly used, e.g. Baluja et al. (2012), Yang and Everitt (2012) and Hall et al. (2011). These are recent examples from the PA Journal. They are not chosen to specifically criticise the PA Journal or these particular publications, but to provide some examples from this journal. Many other examples of an uncorrected (conventional) Pearson s correlation applied to spatial data were found from earlier years and in other journals that have published PA articles. The concern is that the correct interpretation of Pearson s correlation is not used in PA studies, possibly because many researchers are unaware of it. This concern can be easily rectified by using the correction of Dutilleul (1993) or Clifford et al. (1989), which is available in some statistical software packages. A somewhat more subjective concern is how to reconcile (co-locate) two variables that are measured with varying degrees of accuracy, precision and spatial density. In particular, how spatially dense on-the-go sensor data, that may have a high stochastic effect and/or measurement error for any individual observation, are reconciled with high-quality, manual point observations that are sampled at a much lower spatial resolution. Three broad possibilities exist; (a) interpolating both to a new common grid, (b) interpolating the sensor data to the manual point data locations or (c) interpolating the manual point data onto the sensor locations. In general, option (a) is used in the literature, as the correlation is

3 560 Precision Agric (2013) 14: performed on co-located data generated for map representation. However, the generation of a data-set for the determination of correct correlation coefficients may be a different issue to the generation of a surface for mapping. If high quality data are available, then interpolation will smooth the data and introduce more auto-correlation into the data, thus further decreasing n eff. In contrast, by correctly interpolating noisy on-the-go sensor data, where any particular individual measurement may have an error associated with it, a more coherent spatial response can be generated (Whelan et al. 2001), i.e. correct interpolation can help correct noisy high-spatial resolution sensor data. It would therefore seem likely that in many situations option (b) above would be preferable for correlation analysis (though not necessarily for map generation). It is difficult to envision a situation where option (c) would be preferred. Of course, there are other issues with the use of Pearson s correlation in PA (and any study) that are not related to the sample size and the determination of significance. These include, among others, the understanding that Pearson s correlation is only testing for linear relationships and that a significant result does not imply causation. While these issues are not discussed here, it is always important to consider the suitability and intent of correlation analysis (and any analysis) before performing an analysis. An example To demonstrate these concerns, an example is presented using sensor-derived canopy data and manual pruning weight (vine size) measurements from a vineyard. Individual vine pruning weights (PW) were measured post-harvest in 2011 on vines within a 0.93-ha vineyard of Concord (Vitis Labruscana cv Bailey) juice grapes in Portland, NY, USA. The PW was measured using the method of Jordan et al. (1981) and transformed into kg of pruned wood per meter of cordon wire, based on a vine x row spacing of m. During the 2011 season, the normalised difference vegetation index (NDVI) (Rouse et al. 1973) was also measured on this 0.93-ha block with a GreenSeeker RT100 sensor (Ntech Industries Inc., Ukiah, CA, USA). The sensor was mounted on a four wheel all-terrain vehicle and the point NDVI (canopy vigour) data were collected along each row at 5-Hz driving at *10 kmph. The NDVI values were then averaged to 1-Hz response and georeferenced with a WAAS-corrected GPS receiver. Sensing was performed during veraison (19 August, 2011), a key phenological period during which previous studies have shown that there is a relationship between PW and proximal-sensed canopy vigour (Drissi et al. 2009; Stamatiadis et al. 2010). The point NDVI (canopy vigour) data were collected by side imaging the canopy at 0.8 m above the ground level (*40 % of the cordon wire height). It is noted that NDVI observations are known to saturate at moderate to high leaf areas (Curran 1983). Therefore, imaging along the dense foliage of the top-wire cordon produces a saturated signal and a skewed distribution of NDVI. By imaging the sidecurtain of the canopy, the sensor is able to distinguish between large vines, with a vigorous side-curtain, and smaller vines with a less vigorous and less developed side-curtain. In these sprawl trellis system, this approach inversely mirrors the method used by Drissi et al. (2009) to avoid NDVI saturation in a vertical shoot positioned trellis. Prior to analysis, some data clean-up was performed due to errors in the data collection. The PW values associated with missing or renewal vines (PW \ kg m -1 ) were removed. This was a distinct population in the dataset. Renewing vines is a common practice in this viticultural region, thus a percentage of vines in the vineyard are normally missing or small, i.e. in the process of being returned to a full production potential. While

4 Precision Agric (2013) 14: Fig. 1 Experimental variogram clouds for the a NDVI sensor and b manual pruning weight observations. Both show an increase in semi-variance as the lag (distance between the pairs) increases, which is indicative of auto-correlated data each PW observation was a precise point measurements, there is some stochastic and measurement error in the high-density spatial NDVI observations. Healthy late-season vines should return NDVI values in a range of , depending on the amount and health of the green material being sensed and so the NDVI data were trimmed to this range. The NDVI values outside of this range indicate either error in sensor operation or signal processing during the on-the-go sensing, or an absence of green leaf in the scanning area, e.g. a missing/renewal vine or during turning operations at the ends of rows. The data clean-up for both data-sets therefore constrained the data (and analysis) to observations associated with productive vines. After the data clean-up, there were PW and NDVI (1 Hz) observations. Both data-sets approximated a normal distribution, an assumption for calculating Pearson s correlation (PW: l = 0.45, median = 0.45, range [0.06, 0.99], skewness (g 1 ) = 0.38; NDVI: l = 0.88, median = 0.88, range [0.60, 0.95], g 1 =-0.45). Following the data clean-up, the experimental variogram cloud for both variables was generated using Vesper Ò (Minasny et al. 2005) and plotted (Fig. 1). Both variables

5 562 Precision Agric (2013) 14: Table 1 Correlation analysis between vine pruning weight (PW) (actual and interpolated values) and interpolated values of NDVI from a proximal canopy sensor Variable Conventional (non-spatial) Corrected for spatial processes A B r p n total p (Dutilleul) n eff Vine PW NDVI \ (vines) Kriged PW NDVI \ (grid) The p-values and sample size (n) associated with a conventional (non-spatial) Pearson s correlation analysis are shown as well as the p-values and effective sample size (n eff ) for Pearson s correlation after Dutilleul s correction has been applied to account for spatial processes in the PW and NDVI data exhibited spatial autocorrelation in their respective experimental variogram clouds that should be accounted for when determining the actual number of independent observations in both data-sets. To perform Pearson s correlation these data needed to be co-located. Two approaches to co-locate these data were performed, equivalent to options (a) and (b) discussed previously. The NDVI data were interpolated onto the vine locations using a 5.5-m (29 row width) block-kriging approach. The NDVI and PW data were also interpolated onto a m regular grid (equivalent to row spacing) over the block, again using the 5.5-m (29 row widths) block-kriging approach. All interpolation was done with the Vesper Ò software. A correlation analysis between NDVI and PW at the vine sites and on the regular 2.75-m grid was performed in the PASSaGE (v.2) shareware suite (Rosenberg and Anderson 2011), with and without Dutilleul s correction adjustment for sample size. The correlation results are shown in Table 1. This analysis clearly showed the effect of auto-correlation and interpolation on the statistical significance between the two variables. Without correction (conventional analysis) the correlation was strongly significantly for both approaches (p \ 0.001). This is typical of the p-values given in PA publications that involve correlation analysis on imagery data or interpolated surfaces. The raw PW measurements contained a large amount of vine-to-vine variation (data not shown but evidenced by the large nugget effect (semi-variance at a lag distance of 0.0 m) in Fig. 1b). However, when this was smoothed using the 5.5-m block-kriging, spatial patterns were clearly evident in the resultant maps (not shown). After interpolation, the kriged PW had a larger correlation coefficient than the vine PW with the interpolated NDVI (r = c.f. r = 0.179), although there was no difference in the level of significance between the two approaches when using n. When the spatial processes in the variables were accounted for the significance differed. There was no change in the r-values (the correlation coefficient is always calculated with n data), but there were differences in the interpretation of the significance of the correlations. The n eff for the kriged NDVI correlation with the raw PW decreased from to 174.2, and the p-value increased to Therefore, this correlation was significant at p \ 0.05 but not at p \ In the second example, the block-kriging interpolation of PW had smoothed the short-range variation in the PW and increased the auto-correlation in the PW data. The increased auto-correlation in the kriged PW data decreased the n eff to 16.1 for the correlation analysis between the two kriged variables, even though the total n slightly increased with the grid size (grid was slightly larger than the actual block and the vine data had 45 missing vines (points)). The lower n eff resulted in less significance for the kriged PW and NDVI correlation c.f. vine PW and kriged NDVI correlation, even though the r-value was

6 Precision Agric (2013) 14: stronger. The kriged PW and NDVI correlations were statistically significant at p \ 0.1 but not at p \ The results in Table 1 illustrate the care needed in interpreting the significance associated with Pearson s correlation on auto-correlated spatial data. As ever, correlation analysis remains an indicative analysis and the interpretation is dependent on the quality of the data, the situation to which it is being applied and the expected strength of relationships. Correlation coefficients and significance that are derived from data collected within agricultural production systems require an agronomic as well as statistical interpretation. In this regard, the results and discussion from many (if not all) of the articles found that are based on an uncorrected Pearson s correlation are probably valid; however the statistical analysis used is flawed. After correcting for the auto-correlation in the data, some pairwise correlations may present considerably less significance. How a scientist or grower interprets a hypothetical r-value of 0.3 will always depend on the circumstance; but it is likely to be a different interpretation if the associated p-value is or 0.5. From a scientific perspective, if statistical significance measures are to be attached to a correlation analysis then these should certainly take into account any auto-correlation in the data. Growers and industry end-users are more likely to consider the r-value without considering the p-value (statistical significance). In this case, how the data are presented is important. In Table 1 the relationship between vine PW and kriged NDVI may be dismissed as not important (r = 0.179), whilst the relationship between the kriged PW and kriged NDVI may be considered of importance (r = 0.471). The r-value presented may change the decision a grower or consultant makes in regards to adopting or further testing canopy sensors for estimating vine size in this system; even though, from a statistical perspective, the lower r-value (vine PW and kriged NDVI) had a lower p-value and held more statistical significance. The authors envision that spatial data are likely to become more prevalent in agricultural decision-making and skills in analysing and interpreting spatial data more important in the coming decades. This will not be limited to correlation analysis, the focus of this brief discussion article. The increased availability of spatial data in the environmental and agricultural domains has led to several useful review articles on the need to account for spatial auto-correlation in spatial data analysis over the past few years (e.g. Dormann et al. 2007; Beale et al. 2010; among others). It is also important to note that there are many excellent articles in the PA domain that deal with issues associated with auto-correlated data in modelling spatial agricultural responses. Hopefully this discussion will help ensure that the same is achieved with non-spatial descriptive statistical approaches. Acknowledgments The authors would like to acknowledge the diligence and efforts of the technical staff at CLEREL during the collection of the NDVI and pruning weight data. References Baluja, J., Diago, M., Goovaerts, P., & Tardaguila, J. (2012). Assessment of the spatial variability of anthocyanins in grapes using a fluorescence sensor: Relationships with vine vigour and yield. Precision Agriculture, 13(4), doi: /s x. Beale, C. M., Lennon, J. J., Yearsley, J. M., Brewer, M. J., & Elston, D. A. (2010). Regression analysis of spatial data. Ecology Letters, 13, Clifford, P., Richardson, S., & Hemon, D. (1989). Testing the association between two spatial processes. Biometrics, 45, 134.

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