Effect of Selected Factors on the Prediction Accuracy of Plant-Available Phosphorus and Potassium: A Global Meta-Analysis for Infrared Spectroscopy Protocol

Abstract

Chemical methods are reliable for soil testing, but they can be both time-consuming and expensive. This reason makes them unsuitable for quick analysis, especially for many soil samples. Infrared spectroscopy is a cutting-edge technique that offers promising alternative methods that are fast, cheaper, and environmentally friendly. Research findings show contradicting accuracy levels for the infrared protocol when predicting phosphorus (P) and potassium (K). For this reason, the study employed meta-analysis to investigate the effect of selected factors on the prediction accuracy of P and K. Studies that tested P and K in the laboratory using the protocol were selected. The results showed that concentration and soil sample size can significantly (p < 0.001) affect the prediction accuracy of these plant-available nutrients. A positive correlation was observed between the coefficient of determination (r²), sample size (r² = 0.75 P, r² = 0.68 K), and concentration (r² = 0.62 P, r² = 0.64 K). The variation observed in the dependent variable (r²) was explained mostly by sample size and concentration. This could be the source of the overall low prediction accuracy observed from the studies [r² = 0.46, RPD = 1.44 (P) and r² = 0.55, RPD = 1.55 (K)].

1. Introduction

Measurement of plant-available phosphorus (P) and potassium (K) is important for efficient and adequate fertilizer application to achieve optimum crop growth [1]. Soil nutrient measurement can be conducted using either chemical methods or infrared spectroscopy. Chemical methods are reliable but expensive and time-consuming when analyzing large numbers of soil samples. In contrast, infrared spectroscopy offers quick, cheap, and environmentally friendly alternatives [2]. However, the accuracy of the infrared spectroscopy protocol still needs to be investigated, especially for plant-available P and K.

Accuracy in spectroscopy refers to how close results are to the actual values (reference data), while precision is the degree of reproducibility of the results, regardless of accuracy [3]. The accuracy of an infrared spectroscopy protocol is mainly assessed in terms of overall accuracy and precision using validation tests or statistics from a particular or selected regression model. The validation results of the regression model typically include the coefficient of determination (r²), the ratio of performance to deviation (RPD), and the root-mean-square error (RMSE). The r² measures the proportion of the total variation explained by the model; the RPD is the ratio of the standard deviation of the reference data to the RMSE in the calibration or validation dataset [3,4]. The RMSE quantifies the difference between model-predicted and measured values [5].

Researchers have established assessment standards for the overall performance of a model and for ensuring accurate and precise results when analyzing soil samples [6,7,8]. The main benchmarks include, but are not limited to, r² > 0.9 and RPD > 3 (very reliable); r² = 0.7–0.9 and RPD = 1.75–3 (reliable); and r² < 0.7 and RPD < 1.75 (less reliable). The validation results for phosphorus and potassium measured using infrared spectroscopy are generally classified as less reliable in most reported studies. For P, the coefficient of determination obtained from various soils [8,9,10,11,12,13,14,15] is below the recommended threshold (r² < 0.64–0.0) in most cases. Similarly, K validation results show comparable trends (r² < 0.65–0.0) [16,17,18,19,20,21,22].

It has been noted that poor P and K predictions can be attributed to factors related to sample preparation, spectrum acquisition, spectral pre-treatment, soil texture, geological heterogeneity, reference methods, calibration methods, and instrument type [23]. However, many of these factors can be controlled to improve prediction accuracy. For instance, a portion of studies have achieved reliable validation results (r² > 0.70) for plant-available phosphorus [24,25] and potassium [26,27,28] despite these challenges.

The accuracy of reference data (concentrations) is the most limiting factor, whereas other aspects, such as the quality of soil spectral data, can be improved using powerful computer software [3]. Research findings indicate that the concentration of soil properties can significantly influence prediction accuracy [29]. For example, soil samples with a wide range (0–120%) of P and K concentrations were tested, and the results showed inconsistent predictions for both P and K, indicating that concentration is a major factor affecting prediction accuracy when using the protocol [30]. Concentration is measured from a variety of soils or soil sample sizes, typically from 10% of the total soil sample [31]. A limited number of studies have investigated the effect of concentration and sample size on the prediction accuracy of P and K when using the infrared spectroscopy protocol. Therefore, the current study aimed to investigate the poor prediction accuracy of P and K in soils by examining the effects of P and K concentration and soil sample size on prediction accuracy using a global meta-analysis.

2. Methodology

2.1. Meta-Analysis

Meta-analysis is a quantitative method for comparing and linking results from independent studies in order to estimate treatment effects and variability, and identify patterns among study results [32]. This study used meta-analysis to analyze the effect of P and K concentration and soil sample size on the prediction accuracy (r²) of infrared spectroscopy protocol. Infrared spectroscopy is generally a measure of the interaction between light and a sample [33], following the steps displayed in Figure 1.

2.2. Literature Search Strategy

A systematic literature search was conducted manually to identify peer-reviewed studies that used infrared spectroscopy to predict soil phosphorus (P) and potassium (K) in the laboratory (ex situ), using soil samples collected in the field or archive soil samples (spectral library). Electronic searches of peer-reviewed articles were carried out in Google Scholar, ScienceDirect and ResearchGate across the globe (Figure 2) using keywords such as

• Infrared spectroscopy and soil nutrients;
• Spectroscopy and soil fertility;
• Prediction of soil properties;
• Visible near-infrared, near-infrared, mid-infrared and soil P and K;
• Calibration models and prediction of soil properties.

2.3. Selection Criteria

Selection of studies was not limited to a year or geographic location. Studies were selected if they met the following criteria:

• Used infrared spectroscopy to predict soil properties in the laboratory;
• Reported reference chemical methods and soil analysis results;
• Reported validation prediction statistics (r², RMSE, and RDP);
• Predicted plant-available phosphorus and potassium;
• Provided soil sample sizes;
• Conducted experimental research.

2.4. Data Extraction

Data was collected from the Abstract, Materials and Methods, Results, and Discussion sections of each article. For each selected study the following data was extracted:

• Author names, year of publication, place of research (country);
• Treatment means for P and K measured using chemical methods;
• Soil sample size;
• Validation statistics (r², RMSE, and RPD) for either P or K;
• Type and name of instruments;
• Chemical methods;
• Infrared regions;
• Regression models.

2.5. Data Sorting

Data were captured and sorted in Microsoft Excel. Treatment means for P and K concentration were uniformly standardized to milligrams per kilogram of soil (mg/kg). Thereafter, the coefficient of determination (r²), RPD, and RMSE values were linked to the soil sample size and nutrient concentration for each study. The entire dataset was arranged using custom sorting, with concentration ordered from the smallest to the largest value for each nutrient. The data were cleaned by removing duplicates and correcting spelling errors in the meta-data (countries, instruments, chemical methods, infrared regions, and regression models).

2.6. Statistical Analysis

Microsoft Excel, IBM SPSS Statistics version 20, and Jamovi version 2.5 were used for data analysis in this study. Microsoft Excel was used to clean, sort, and analyze the data (Figure 2), as well as to examine the relationships among r², RPD, and RMSE (Figures 7 and 8). The meta-data (

Table 1. A summary of chemical methods, infrared regions, models, and spectrometers used in the selected studies.

Infrared Region	P (%)	K (%)	Total (%)
Visible near-infrared	20.5	17.9	38.4
Near-infrared	10.3	13.7	24
Mid-infrared	18.8	18.8	37.6
Spectrometer
Carry 500	1.7	1.7	3.4
NIRS 6500	4.3	3.4	7.8
FT-IR-Spectrometer	19.8	21.6	41.4
ASD Field Spec	12.9	12.1	25.0
NIR portable spectrometer	1.7	0.0	1.7
XDS rapid content analyzer	4.3	4.3	8.6
Foss NIRS 5000	0.9	3.4	4.3
Neo spectra	2.6	3.4	6.0
NIR-M-R2	0.9	0.9	1.7
Regression Model
Principal component regression	0.9	2.2	3.1
Partial least squares	34.2	32.5	66.7
Multiple linear regression	5.1	2.1	7.2
Convolutional neural network	2.6	3.4	6
Cubist	3.4	1.7	5.1
Artificial neural networks	1.7	1.7	3.4
Support vector regression	3.4	2.6	6.0
Multivariate adaptive regression splines	0.0	0.9	0.9
Chemical method
Mehlich 1	5.8	1.2	7.0
Mehlich 3	19.8	15.1	34.9
Ammonium fluoride	1.2	0.0	1.2
Ammonium acetate	1.2	15.1	16.3
Ammonium oxalate	2.3	0.0	2.3
Calcium acetate	0.0	2.3	2.3
Ammonium chloride	0.0	3.5	3.5
Bray-P	8.1	0.0	8.1
Olsen-P	12.8	0.0	12.8
Resins	2.3	0.0	2.3
Silver-thiourea	0.0	2.3	2.3
Water extraction	1.2	0.0	1.2
Bicarbonate extractable	3.5	1.2	4.7
Lancaster method	1.2	0.0	1.2

) was analyzed using frequency distributions (%) in IBM SPSS Statistics version 20. Jamovi version 2.5 was used to analyze the correlation between prediction accuracy (r²), soil sample size and concentration generated from the selected studies. The software was used not only for correlation analysis but also for data reduction. Some studies were identified as non-influential, contained repeated values, and these were removed from the analysis. Influential studies were identified by evaluating changes in the r-estimate or the Fisher r-to-z–transformed correlation coefficient Equation (1) when individual studies were excluded. Also, the model fails to run when finite xlim values are present in the dataset.

$$\mathrm{Z}={\displaystyle \frac{1}{2}} \mathrm{I}\mathrm{n}\left({\displaystyle \frac{1+\mathrm{r}}{1-\mathrm{r}}}\right)$$

(1)

Equation (1) Fisher r-to-z transformed correlation coefficient.

The Fisher r-to-z transformation corrects non-normally distributed correlation coefficients by converting them into values that follow an approximately normal distribution. Its primary purpose is to normalize the distribution of correlation coefficients and stabilize their variances, thereby making meta-analytic methods more robust and reliable [35]. A random-effects model was fitted to the data, and the correlation effect size was selected. The studies were weighed against the overall mean for sample size and concentration for both P and K. Significant differences were determined using the Z-score, which measures the distance from the mean, where Z = 0 indicates no significant difference.

3. Results and Discussion

3.1. Meta-Data

Relevant or selected articles were published between 2001 and 2024. The number of publications increased steadily over this period (Figure 2), with Africa recording the lowest number of studies. Most publications investigated multiple factors such as sample size, infrared regions, models, and chemical methods within a single study. A total of 118 experiments were extracted for the meta-analysis, including 58 for phosphorus and 60 for potassium (

Table A1. The studies used to extract meta-data and experimental data for analysis.

NO.	Year of Publication	Study Reference
1.	2001	[16]
2.	2003	[4]
3.	2003	[40]
4.	2003	[41]
5.	2005	[38]
6.	2006	[9]
7.	2007	[27]
8.	2007	[11]
9.	2007	[7]
10.	2008	[42]
11.	2008	[26]
12.	2009	[12]
13.	2009	[43]
14.	2010	[44]
15.	2010	[45]
16.	2012	[18]
17.	2012	[19]
18.	2012	[46]
19.	2013	[47]
20.	2014	[48]
21.	2014	[49]
22.	2014	[50]
23.	2015	[28]
24.	2015	[51]
25.	2016	[52]
26.	2017	[53]
27.	2018	[14]
28.	2018	[54]
29.	2019	[15]
30.	2019	[55]
31.	2020	[25]
32.	2020	[21]
33.	2020	[22]
34.	2020	[56]
35.	2021	[57]
36.	2021	[58]
37.	2021	[59]
38.	2022	[60]
39.	2022	[61]
40.	2023	[62]
41.	2023	[63]
42.	2024	[64]
43.	2024	[65]
44.	2024	[66]
45.	2024	[67]

). The selected studies measured soil P and K using infrared spectroscopy protocols (Figure 1) in the laboratory or ex situ. Soil samples were air-dried and finely ground to less than 2 mm to eliminate moisture and texture as confounding factors [14]. Soil spectra were measured in the visible near- or mid-infrared regions, mainly using Fourier transform infrared spectrometers.

Vis–NIR contributed 20.5% of P and 17.9% of K predictions, while MIR contributed 18.8% for both P and K. The slightly higher Vis-NIR is mostly a contribution from older studies. In recent years, researchers are shifting to MIR because of its known advantage in capturing fundamental vibrational absorptions associated with mineral and organic functional groups, which are often strongly correlated with nutrient availability. In contrast, the lower contribution of NIR-only studies (24%) suggests that NIR alone is less frequently adopted, likely because it captures weaker overtones and combination bands. The most frequently used spectrometers were FT-IR instruments, contributing 41.4% of all data, with similar shares for P (19.8%) and K (21.6%). This supports the trend towards MIR-based spectroscopy, as FT-IR devices are commonly used for MIR measurements. The ASD Field Spec instruments represent 25.0% of the data, indicating a lower value when compared to FT-IR instruments. ASD can be used in the field or laboratory (portable Vis–NIR instruments). The scope of the search focused more on laboratory studies, which limited studies that used field spectrometers for plant-available P and K.

Partial least squares regression was the primary modeling approach, applied in 66.7% of the studies to analyze soil spectral data for P and K. Reference data were obtained mainly using the Mehlich-3 method (34.9%). However, the variation in reference methods for P Mehlich 3 (19.8%), Olsen-P (12.8%), and Bray-P (8.1%), and K ammonium acetate extraction (15.1%), and Mehlich 3 (15.1%) demonstrates the lack of global standardization for soil P and K testing. The results for meta-data, which includes infrared regions, spectrometers, regression models, and chemical reference methods are summarized in Table 1.

3.2. Effect of Soil Sample Size and Concentration on the Prediction Accuracy of P and K

Table 2. Effect of sample size and concentration on the prediction accuracy of P and K.

Statistics	Sample Size			Concentration (mg/kg Soil)
	P	K	Average	P	K	Average
r	75%	68%	72%	62%	64%	63%
Z	7.48	10.8	9.14	6.06	9.84	6.45
Heterogeneity (p-value)	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001
RE model (p-value)	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001

summarizes the meta-analytic statistics for phosphorus and potassium prediction performance after fitting the random-effects (RE) model, allowing comparison of the effect of sample size and soil nutrient concentration on the prediction accuracy of the infrared protocol. The table presents the pooled correlation coefficients (r), Fisher’s z-transformed effect sizes (Z), and the corresponding heterogeneity statistics used to evaluate the variability among studies.

The statistical analysis at the 95% confidence interval (CI) indicated a highly significant RE model and substantial heterogeneity (p < 0.001). These results demonstrate that both sample size and nutrient concentration significantly influence the prediction accuracy of plant-available phosphorus and potassium (Table 2). The significant heterogeneity suggests that the included studies were conducted under diverse conditions, such as differences in soil types, analytical instruments, and methodological approaches. This is further supported by the Z-scores, all of which were greater than zero, indicating meaningful differences among studies.

The pooled prediction accuracy was higher for P (75%) than that for K (68%), with a combined average accuracy of 72%, suggesting that P prediction is more sensitive to sample size and related soil characteristics. A positive effect was also observed for nutrient concentration; however, accuracy values were relatively similar for P (62%) and K (64%). Overall, sample size emerged as the most influential factor affecting prediction accuracy for both nutrients, yielding the highest average accuracy (73%) compared with nutrient concentration (63%).

3.2.1. Soil Sample Size

The results presented in Section 3.2 are summarized using forest plots shown in Figure 3, Figure 4, Figure 5 and Figure 6. Each forest plot is divided into three components: the study reference, the line of no effect, and the correlation coefficients (r) accompanied by their 95% confidence intervals, CI-lower and CI-upper limits. Phosphorus (P) sample sizes ranged from 35 to 660, while potassium (K) sample sizes ranged from 48 to 908. Based on these ranges, twenty-one (21) studies for P and twenty-six (26) studies for K were included in the generation of the forest plots. Overall, the forest plots revealed a positive effect of soil sample size on prediction accuracy. The results demonstrating the influence of sample size on the prediction accuracy for P and K are presented in Figure 3 and Figure 4.

The heterogeneity observed for sample size was consistent with the statistical analysis presented in Figure 3 and Figure 4 and was further supported by the results in Table 2. The effect of sample size on prediction accuracy was significant and varied across studies, with the exception of two studies whose confidence intervals overlapped the line of no effect, indicating non-significance. The variation observed among the studies was expected and aligns with published empirical findings [36]. Soil samples are inherently variable in their physical and chemical properties. For example, soil color can indicate the presence of organic matter, iron oxides, and carbonates, all of which influence light reflectance and can ultimately affect prediction accuracy [37]. In addition, the size, shape, and arrangement of soil particles can influence light transmission paths and the resulting spectra [16]. The number of soil samples is therefore important to ensure that the full range of soil variability is captured by the regression model, leading to more reliable and accurate predictions. However, larger sample sizes do not always translate into higher prediction accuracy. Some studies with relatively large sample sizes still reported poor prediction performance for both P and K, indicating that other factors such as data quality, spectral pre-processing, model choice, and soil heterogeneity also play critical roles.

3.2.2. Soil Nutrient Concentration

Soil nutrient concentrations varied widely across the studies, ranging from 8.4 to 215 mg/kg for phosphorus (P) and from 15.64 to 600 mg/kg for potassium (K). A total of twenty-four (24) studies for P and twenty-seven (27) studies for K were included in the forest plot analysis. The results were consistent with the statistical findings presented in Table 2, showing that nutrient concentration significantly affected prediction accuracy for both P and K (p < 0.001; see Figure 5 and Figure 6).

Approximately ten (10) P studies showed no significant effect, as indicated by error bars overlapping the line of no effect. These studies also had small square sizes and wide confidence intervals, reflecting their limited influence on the overall effect size. Despite the presence of these non-significant studies, phosphorus concentration still explained 62% of the variation in prediction accuracy.

For potassium, only four (4) studies showed no effect (Figure 6), which aligns with the slightly higher correlation coefficient (64%) observed for K. Significant heterogeneity (p < 0.001) was detected among studies for both nutrients, reflecting natural variability in soil concentration levels. Such variability is expected because soil nutrient concentrations are strongly influenced by local agronomic practices, which differ across regions and management practices.

3.3. Accuracy of the Infrared Spectroscopy Protocol

The study evaluated the effectiveness of the infrared protocol for predicting soil phosphorus (P) and potassium (K) concentrations using the ratio of performance to deviation (RPD). RPD quantifies model performance by comparing the standard deviation of reference (chemically measured) values with the root mean square error (RMSE) of predictions. The coefficient of determination values (r²) reported in earlier sections were analyzed alongside the corresponding RPD and RMSE values. However, some publications did not report RPD and RMSE; only a subset of the extracted studies could be used to assess the predictive accuracy of the protocol for P and K.

Chemical methods are regarded as the true or reference values for nutrient concentrations [38]. Therefore, predictions that are closer to the actual values are known to be accurate. The closeness of measured values (chemical methods) and predicted values (models) can be estimated using both RMSE and RPD. A smaller RMSE and RPD > 2 represent a reliable or accurate model [39]. The observed average RPD recorded from the studies was below the standard (<2 RPD) for a reliable regression model [39]. In this study, the average RPD values obtained from the reviewed publications were below this threshold (<2), indicating weak model performance. The regression models produced RPD values of 1.41 for P and 1.55 for K, suggesting that the protocol predicted K more accurately than P. The overall prediction performance was r² = 0.46 for P and r² = 0.55 for K.

As established in the preceding sections, soil sample size and nutrient concentration were the primary drivers of the observed low prediction accuracy. The poor predictive performance for P and K can be further attributed to their relatively low concentrations in soils [28], as well as differences in nutrient mobility within the soil solution, both of which influence spectral detectability. The correlation analyses among r², RPD, and RMSE are presented in Figure 7 and Figure 8.

3.3.1. Phosphorus

Phosphorus reference data (chemical methods) were extracted mainly using Olsen-P, Bray-P, Mehlich 1, and Mehlich 3 (Table 1). Accurate P predictions are most likely to be observed from the Mehlich 3 and Olsen-P methods [16,24]. The results showed a positive linear relationship (Figure 7a) between the r² and RPD values [12] and a negative relationship (Figure 7b) between r² and RMSE. An increase in the phosphorus RPD (P-RPD) values corresponded to an increase in the coefficient of determination, while the inverse was observed for the phosphorus RMSE (P-RMSE). The observed error (RMSE) decreased as the prediction performance (r²) of the protocol increased, hence the negative correlation (Figure 7b). Increasing the number of soil samples can reduce the observed error or RMSE [4]. Despite the relatively low predictability of P, the positive correlation between r² and RPD values, which reflects the agreement between measured and predicted results, indicates that the protocol is accurate. However, further investigation is required to explore the effect of P concentration levels on the prediction performance of the protocol.

3.3.2. Potassium

The reference data for potassium were extracted mainly using ammonium acetate and Mehlich 3 (Table 1). Reference data obtained using Mehlich 3 generally provided more accurate predictions than those extracted using ammonium acetate [16]. The coefficient of determination (r²) from the validation results was correlated with the potassium RPD (K-RPD) and potassium RMSE (K-RMSE) values. The correlation patterns were similar to those observed for phosphorus (Section 3.3.1). A positive linear relationship between r² and K-RPD was observed (Figure 8a), indicating that K-RPD increased proportionally with prediction performance. Likewise, a downward trend was detected in the K-RMSE results (Figure 8b), where prediction accuracy decreased as RMSE increased, as expected.

Overall, the protocol demonstrated higher accuracy for predicting potassium than phosphorus. A strong relationship (r² = 0.79) was found between K-RPD and r², compared with the weaker association for phosphorus (P-RPD; r² = 0.46). These results align with the findings of [27], who also reported more accurate predictions for K (r² = 0.68) than for P (r² = 0.46). Collectively, the findings confirm that the protocol is reliable for predicting both P and K. However, further refinement and a deeper understanding of the protocol are required to improve prediction accuracy, particularly for phosphorus.

4. Conclusions

This study identified two key factors, sample size and nutrient concentration, that substantially influence the accuracy of predicting plant-available phosphorus (P) and potassium (K) using infrared spectroscopy. The results demonstrate that the performance of the protocol is sensitive to the diversity and representation of soil samples used to build the spectral library. Although larger sample sizes broaden the range of soil variability, they did not consistently yield higher prediction accuracy, indicating that sample quality and representativeness may be more critical than quantity alone.

Nutrient concentration also emerged as an important factor influencing prediction accuracy. In cropland systems, nutrient levels are difficult to control due to continuous crop uptake, inherent soil variability, and fluctuations in climatic conditions. As a result, soil nutrient concentrations are often highly variable. This variability contributed to a higher number of studies showing no significant effect in the forest plots for concentration compared to sample size. The findings therefore highlight a research gap regarding the role of P and K concentration levels in shaping the predictive performance of infrared spectroscopy. Overall, the protocol appears to reliably measure what it is intended to measure. The positive relationship observed between r² and RPD indicates a meaningful correspondence between chemically measured and model-predicted values, confirming that the infrared protocol can effectively predict P and K concentrations.