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Controversy exists on the magnitude and variability of farm nutrient balances and quality of arable land in sub-Saharan Africa with Kenya among those affected negatively. This study investigates quality of arable land by fitting multivariate multilevel model to farm nutrient balance data collected from five agro-climatic zones of Kenya (arable lands). Objectives of the study were to investigate the magnitude and variability of Nitrogen, Phosphorus and Potassium (NPK) farm nutrient balances in arable lands of Kenya, study effects of agro-climatic zones on nutrient balances and to determine effects of household resource endowments on NPK nutrient balances. The study concludes that agro-climatic zones differ with respect to farm nutrient balances; that livestock resource endowments and hired labour have positive effects on the magnitude and direction of farm nutrient balances; and that household ownership of large capital resources do not guarantee a positive effect on farm nutrient balances. The study recommends integration of sound livestock practices and application of agro-climatic zone differentiated interventions in future strategies for addressing farm nutrient balances and arable land quality, and the use of large sample sizes and relevant factors/ covariates in future analysis to shed additional insights on farm nutrient balances and on how arable land quality can be im proved.

Seminal studies conducted at national and regional levels in sub-Saharan Africa, using nutrient balance approach, have indicated declining arable land quality with severe net nutrient losses of the order of 10 kg Nitrogen, 4 kg phosphates and 10 kg potash per hectare annually [

One of the reasons why consensual accounts on nutrient balances remain intractable and illusive and at times anecdotal [

Approaches to model estimation in meta-analysis vary widely. Descriptive analysis and paired t-test have been used in meta-analysis of nutrient balances drawn from 57 studies in Africa and concluded that there were positive soil nitrogen and potassium balances in some spots in Africa [

Current methods of meta-analysis, however, have several limitations [

Possible approaches to modeling multiple outcomes of nutrient balances, taking into account the above challenges, include: multivariate fixed and random effects models, structural equation models, and multilevel models for modeling primary data among others [

In this study we demonstrate that multivariate multilevel models can be used in meta-analysis of farm nutrient balance data arising from complex surveys that involve multi-stage sampling, stratification and unequal sampling probabilities [

This study uses individual participant data from multiple related cross-sectional surveys on nutrient balances from five different agro-ecological zones of Kenya to investigate the quality of arable land by estimating the magnitude and variability of Nitrogen, Phosphorus and Potassium (NPK) nutrient balances, assessing whether agro-climatic zones differ with respect to NPK nutrient balances and determining the effects of household resource endowments on NPK nutrient balances. To meet the research objectives, the study fitted a two-level multilevel model (multivariate multilevel model) with random intercept to farm nutrient balance data in a meta-analysis that used Iterative Generalised Least Squares (IGLS), an equivalent maximum likelihood method [

NUTrient MONitoring (NUTMON) data is used in this study. NUTMON is part of on-going research to investigate land quality and sustainability of smallholder farming systems in the tropics. The data used comprise 14 separate studies, from 5 research initiatives that used NUTMON methodology in different agro-climatic zones of Kenya. A single research initiative working in “n” agro-climatic zones was considered to have “n” separate studies (where n = number of studies). Studies which did not use multi-stage sampling to identify study participants were excluded from the analysis (on-farm and on-station experiments excluded).

The data comprised 349 observations (individual smallholder farm-households). About 42% and 25% of the smallholder farm-households in the dataset were from semi-humid to semi-arid (ACZ4), and semi-arid areas (ACZ5) of Kenya respectively. Farm households from humid (ACZ1), sub-humid (ACZ2) and semi-humid (ACZ3) areas accounted for 12%, 15% and 7% of total households in the dataset respectively. The arid (ACZ 6) and very arid (ACZ 7), with very low potential for plant production, were not represented in the dataset (

The 349 observations have 3 dependent variables: N full balance (kg ha^{−}^{1}); P full balance (kg ha^{−}^{1}); and K full balance (kg ha^{−}^{1}) and 18 selected independent variables (factors/covariates). The latter were measured at two levels: 1) at level of individual farmers (household resource endowments); and 2) at agro-climatic zone level (

The study used the following general analysis methods:

1) Determined whether a two-level multi-level model with multiple outcome variables (multivariate multi-level model) is required for the NUTMON dataset.

Study acronym | Humid (ACZ1) | Sub-humid (ACZ2) | Semi-humid (ACZ3) | Semi-humid to semi-arid (ACZ4) | Semi-arid (ACZ5) | Total |
---|---|---|---|---|---|---|

ENSET INMASP LEINUTS NUTSAL VARINUTS Total | 0 0 36 0 6 42 | 0 46 0 0 6 52 | 18 0 0 0 6 24 | 9 59 0 71 6 145 | 9 0 35 36 6 86 | 36 105 71 107 30 349 |

Description | Number of variables | Explanations | |
---|---|---|---|

1A | Level 1 factors/covariates | ||

1.1 | Household Resource endowment | 17 | Comprise labour, land units, livestock, nutrient stocks and crop and livestock diversity |

2 | Level 2 | ||

2.1 | Agro-climatic zone (ACZ) | 1 | ACZ1 (Humid), ACZ2 (Sub-humid), ACZ3 (Semi-humid), ACZ4 (Semi-humid-to-Semi arid), ACZ5(Semi-arid) |

Total variables | 18 |

2) Based on (1) above, applied a two-level multi-level model (multivariate multi-level model) to:

Estimate an aggregate magnitude and variability of nutrient balances across agro-climatic zones that cover arable lands of Kenya;

Determine whether agro-climatic zones differ from each other in terms of NPK nutrient balances; and to

Identify the effects of household resource endowments on NPK nutrient balances.

The study fitted a two-level multilevel model (multivariate multilevel model) without predictors (variance component model) to NUTMON dataset to determine whether multilevel modeling was needed at all for this dataset. Intra-class correlation Coefficient (ICC) and Design effect were calculated to aid in model output interpretation.

The multilevel equations for the variance component model were specified as follows:

Level 1: y i j = β o j + ϵ i j

Level 2: β o j = γ o o + ϵ o j

Written in (mixed model) form by substitution of the level-2 equation into the level-1 equation, the model is:

y i j = γ o o + ϵ o j + ϵ i j (1)

where:

y i j = Individual response variable for i^{th} farmer (level-1) in j^{th} agro-climatic zone;

β o j = Random intercept for j^{th} agro-climatic zone (mean of all individual farmers in j^{th} agroclimatic zone)

γ o o = Random intercept for all j agro-climatic zones (grand mean of all js)

ϵ i j = Residual effect (variation) for i^{th} farmer around the mean of j^{th} agroclimatic zone (random effect)

ϵ o j = Residual effect (variation) for j^{th} agro-climatic zone around the grand mean (of all agro-climatic zones ie across all js)

ϵ i j ~ N ( 0 , σ e 2 ) ; σ e 2 is the variance at individual farmer (level-1)

ϵ o j ~ N ( 0 , σ u 2 ) ; σ u 2 is the variance at agro-climatic zone (level-2)

The study used Iterative Generalised Least Squares (IGLS) estimation algorithms in R2MLwiN package, to return estimates for random coefficients and their standard errors, estimates for deviance statistics and for variances and covariances for single and two level models (

The study calculated Study Design Effect^{1} for the two level model as follows:

Designeffect = 1 + ( n c − 1 ) ICC

where:

2[ | Fixed part | Coefficient | Std. Err | 95% Confidence Interval | |
---|---|---|---|---|---|

Lower boundary | Upper boundary | ||||

2[ | Nitrogen balance | −11.92 | 17.93 | −47.06 | 23.22 |

Phosphorus balance | 9.85 | 4.56 | 0.92 | 18.78 | |

Potassium balance | 5.47 | 6.29 | −6.85 | 17.79 | |

Deviance statistic | 10,461 | ||||

No. of observations | 349 | ||||

One-level model | Nitrogen balance | 2.57 | 4.37 | −11.14 | 6 |

Phosphorus balance | 9.72 | 1.77 | 6.25 | 13.2 | |

Potassium balance | 6.46 | 3.13 | 0.33 | 12.59 | |

Deviance statistic | 10,657.2 | ||||

No. of observations | 349 |

^{1}Quantifies the effects of violating the assumption of independence on standard error estimates; Multiplier to be applied to standard errors to correct for negative bias that results from nested data.

n c = Average number of farmers per study; In this case (349/14) = 28.1

ICC = Intraclass correlation coefficient (at level 2); an estimate of proportion of variance at level-2

ICC at level-2 was estimated separately for Nitrogen, Phosphorus and Potassium farm nutrient balances using:

ICC = σ u 2 σ e 2 + σ u 2 (2)

where

σ e 2 = Residual variance at level-1

σ u 2 = residual variance at level-2

σ e 2 + σ u 2 = Total variance at level-2

The design effects for each nutrient balance were greater than 2.0 (

To estimate an aggregate magnitude and variability of nutrient balances across agro-climatic zones of Kenya, the study used Equation (1), describing a variance component model. Iterative Generalised Least Squares (IGLS) in R2MLwiN package used to quantify the parameters of the model, returned parameter estimates shown in (

Similarly, variability of nutrient balances (heterogeneity) at level-1 and at level- 2 model hierarchy were estimated using Variance Partitioning Coeffcient (VPC)/Intra-class correlation coeffcient (ICC) using Equation (2) (see explanation in Section 2.3):

VPC e ( ICC ) = σ e 2 σ e 2 + σ u 2 = var ( ϵ i j ) var ( σ e 2 ) + var ( σ u 2 ) (3)

where

ϵ i j ~ N ( 0 , σ e 2 ) ; σ e 2 is the variance at individual farmer (level-1)

ϵ o j ~ N ( 0 , σ u 2 ) ; σ u 2 is the variance at agro-climatic zone (level-2)

Nutrient balance | n c | ICC | Design effect |
---|---|---|---|

Nitrogen | 28.1 | 0.48 | 14 |

Phosphorus | 28.1 | 0.21 | 6.7 |

Potassium | 28.1 | 0.11 | 4 |

n c = Average no. of farmers per study; ICC = Intraclass correlation Coeffcient.

To determine whether agro-climatic zones differ from each other with respect to nutrient balances, Equation (1) describing a variance component model was used. Parameter estimates were obtained in a similar way as in Section 2.3. The parameter estimates are presented in

Further, in assessing whether agro-climatic zones differ from each other, the study determined whether the variance ( σ u 2 ) of the random component of the intercept in Equation (1), ϵ o j , was different from zero. A 95% confidence interval for the variance of ϵ o j was used to aid model output interpretation. Also, a likelihood ratio test was conducted by comparing the deviances of a model with ϵ o j and one without ϵ o j to assess whether σ u 2 (variance at level-2: agroclimatic zone) is significant. The null hypothesis for the latter was σ u 2 = 0 , so we do not need ϵ o j in the model (Ho: no agro-climatic zone variation or cluster effect exists and restricted or single model is “the true model”).

Natural log used in Likelihood ratio test:

L R = ( − 2 log L 0 ) − ( − 2 log L 1 ) = D 0 − D 1 with 1 df

where:

L 0 = Likelihoodvalueforasinglelevelmodeliewithout ϵ o j

L 1 = Likelihoodvalueforatwo-levelmodeliewith ϵ o j

D 0 = Deviancestatisticsforasinglelevelmodel-without ϵ o j

D 1 = Deviancestatisticforatwo-levelmodeliewith ϵ o j

The p-value associated with the Likelihood ratio (LR) test statistic was determined from Chi Square distribution (with 1 degree of freedom).

The study fitted a two-level multilevel model (multivariate multilevel model) to NUTMON dataset to determine the effects of household resource endowments on NPK nutrient balances. The household resource endowments in

The multilevel equations for this model was specified as follows:

Level 1: y i j = β o j + β k j _ X k i j _ + ϵ i j , (for i = 1 , 2 , 3 , ⋯ , 349 ; j = 1 , 2 , 3 , 4 , 5 ; k = 1 , 2 , 3 , ⋯ , 18 )

Level 2: β o j = γ o o + γ o n _ Z o n j _ + ϵ o j ; β k j _ = γ k o _

Written in mixed model form by substitution of the level-2 equations into the level-1 equation, the model is:

y i j = γ o o + γ o n _ Z o n j _ + γ k o _ X k i j _ + ϵ o j + ϵ i j (3)

where at level-1:

y i j = Individual response variable for i^{th} farmer (level-1) in j^{th} agro-climatic zone;

β o j = Random intercept for j^{th} agro-climatic zone (mean of all individual farmers in j^{th} agroclimatic zone); each agro-climatic zone is assumed to have a different intercept coeffcient, β o j

x k i j _ = A vector of k predictor variables for the i^{th} farmer in j^{th} agro-climatic zone

β k j _ = A vector of k regression coeffcients associated with the predictor variables in j^{th} agro-climatic zone:

x k i j _ = ( X 1 i j X 2 i j X 3 i j ⋮ X k i j ) ; β k j _ = ( β 1 j β 2 j β 3 j ⋮ β k j )

ϵ i j = Residual effect (variation) for i^{th} farmer in j^{th} agro-climatic zone

where at level-2:

γ o o = Random intercept for all five agro-climatic zones (grand mean of all j groups); The ( β o j )s’ are considered to vary randomly around a grand mean of all j groups ( γ o o ) at level-2

Z o n j _ = A vector of n predictor variables measured at agro-climatic zone level (level-2 or j-level)

γ o n = A vector of n regression coefficients associated with the predictor variables at agro-climatic zone level (non-random coefficients):

Z o n j _ = ( Z o 1 j Z o 2 j Z o 3 j ⋮ Z 0 n j ) ; γ o n _ = ( γ o 1 γ o 2 γ o 3 ⋮ γ o n )

ϵ o j = Residual effect (variation) for j^{th} agro-climatic zone; ie the deviation of the intercept of j^{th} agro-climatic zone from overall intercept of all agro-climatic zones (all js)

γ k o = A vector of k (fixed) regression coefficients indicating that the coefficients of Level-1 predictors ( β k j ) do not vary across agro-climatic zone level (non-random slopes at level-2):

β h j _ = ( β 1 j β 2 j β 3 j ⋮ β k j ) = ( γ 10 γ 20 γ 30 ⋮ γ k 0 )

β h j : all j values of β h are fixed (do not vary across agro-climatic zone) and are estimated as a single coefficient γ h 0 at level-2, for h = 1 , 2 , 3 , ⋯ , k ; j = 1 , 2 , 3 , 4 , 5

The two-level multilevel model (multivariate multilevel model) without predictors (variance component model) fitted to the dataset returned the mean NPK nutrient balances, see (^{−}^{1} (with 95% confidence interval: −47.0, 23.2) tended to corroborate results of aggregate seminal studies that have reported negative (direction) nitrogen balances at national level [^{−}^{1}; p < 0.01; 95% Confidence Interval of 0.9, 18.8) and potassium balances (5.5 kg ha^{−}^{1}; 95% Confidence Interval of −6.9, 17.8) were however positive contrary to seminal aggregate studies that reported negative nutrient balances at national level [

The Variance Partitioning coefficients (adjusted Intra-class correlation coeffeicients) for NPK nutrient balances at different levels of model hierarchy are summarised in

For farm nitrogen nutrient balance, 48% of the variation lies between agro- climatic zones (between agro-climatic zone variability) while 52% of variation lie between farms. For each of the nutrient balances studied, a high proportion of total variation was from between-farm variability, 52%, 79% and 89% for nitrogen, phosphorus and potassium respectively.

Based on residual variances of each nutrient balance and covariances at level 1 (farm level;

VPC | Nitrogen balance | Phosphorus balance | Potassium balance |
---|---|---|---|

Level 2 (Agro-climatic zone) Level 1 (Farm) | 0.48 0.52 | 0.21 0.79 | 0.11 0.89 |

Random effects parameters | Coeffients | Standard error | [95% Conf. Interval] | |
---|---|---|---|---|

Agro-climatic zone (Level-2) Var Nitrogen Cov Nitrogen-Phosphorus Var Phosphrus Cov Nitrogen-Potassium Cov Phosphorus-Potassium Var Potassium Farms (Level-1) Var Nitrogen Cov Nitrogen-Phosphorus Var Phosphrus Cov Nitrogen-Potassium Cov Phosphorus-Potassium Var Potassium | 4462.54 779.81 244.42 1175.58 291.61 408.43 4927.10 1689.60 901.30 3215.90 1151.30 3149.10 | 1763.88 392.18 112.36 564.17 144.99 212.66 380.90 147.70 69.70 277.70 111.40 242.70 | 1005.33 11.13 24.19 69.8 7.44 -8.38 4180.50 1400.20 764.80 2671.70 932.80 2673.30 | 7919.74 1548.49 464.65 2281.36 575.79 825.24 5673.70 1979.00 1037.90 3760.10 1369.70 3624.80 |

var = Variance; Cov = Covariance

Nitrogen-potassium (r = 0.87) and Phosphorus-potassium (r = 0.92).

The study assessed whether agro-climatic zones differ from each other, on average, with respect to farm nitrogen, phosphorus and potassium balances. This was explored preliminarily by looking at variance partitioning coefficient (VPC) and two tests i) assessing whether the variance of the random components of the intercept differ from zero and by ii) conducting likelihood ratio test.

Variance partitioning coefficient (VPC) measures the proportion of total variance which lies at the Agro-climatic zone level (level-2). Interpreted as VPC, 48%, 21%, and 11% of variation in nitrogen, phosphorus and potassium farm nutrient balances lie between agro-climatic zones respectively (

Suppose Agro-climatic zones were to differ only slightly or not at all, then the agro-climatic zone j values of ϵ o j (Equation (1)) should differ little from each other and or exhibit low-to-no variance. However, a 95% confidence levels for random part of the model (

The study further used likelihood ratio test to tringulate the observation above as variances are known to have positively skewed sampling distributions while 95% confidence intervals assume assymptotic normal sampling distribution and may not be reliable. A likelihood ratio test done by comparing a model with agroclimatic zone effects (with ϵ o j ) and one without ϵ o j , to assess whether σ u 2 (variance at level-2: agroclimatic zone) is significant returned:

L R = ( − 2 log L 0 ) − ( − 2 log L 1 ) = D 0 − D 1 = 10657 − 10461 = 196.2 with 1 df

The p-value associated with the Likelihood (LR) test statistic (Chi Square value of 196.2) with 1 degree of freedom is 0.0001. Since the p-value is very small, we reject the null hypothesis (Ho: no agro-climatic zone variation or cluster effect exists and restricted or single model is “the true model”) and conclude that a gro-climatic zone variation exists and is significant ( χ ˜ 2 ( 1 , N = 349 ) = 196.2 , p < 0.01 ) . This further confirms that there were significant differences between agro-climatic zones with respect to farm nutrient balances.

A two-level multilevel model (multivariate multilevel model) fitted to the dataset (see Section 4) to test the hypothesis: All household resource endowments do not have an effect on the magnitude of full N, P and K nutrient balances returned results shown in

The observation that more than one household resource endowment variable has an effect on NPK nutrient balances provides a strong evidence against the null hypothesis (e.g. Value of livestock (0.0005 kg N ha^{−}^{1}, p < 0.001); cropping family labour (−0.05101kg K ha^{−}^{1}, p < 0.01),

A negative relationship between family labour for cropping and NPK nutrient balances was observed (

Household resource endowment | Nitrogen (kg ha^{−}^{1}) | Phosphorus (kg ha^{−}^{1}) | Potassium (kg ha^{−}^{1}) |
---|---|---|---|

Constant Average slope% (AVGSLOPE) 20 cm Number of secondary production units (SPUNo: Number of livestock types) Tropical Livestock Units (TLUNo) Value of livestock (VALLVST: in Ksh) Total capital owned (CAPTOT: In Ksh) Cropping family labour (LABCROP: in days) Land rent received (VRENTOUT: in Ksh) Hired labour for RU-Cash (LABHIRUC: in Ksh) | −14.61 −1.57** 9.86** −4.77*” 0.0005*** −0.00001*” −0.03545 0.00001 0.01325 | 16.07** −0.50*” 1.32 −1.35 0.0002** −0.00001**** −0.01307 0.00013*** 0.02318** | 13 −1.61**** 7.36*** −2.65 0.00047**** −0.00001*** −0.05101** −0.00004 0.02078 |

****p < 0.0001; ***p < 0.001; **p < 0.01; *p < 0.05; *”p < 0.1

cant effect (negative) on potassium balance. A unit change in cropping family labour lowering potassium nutrient balance by 0.05101 kg K ha^{−}^{1} (p < 0.01). Although smallholders in Kenya rely heavily on family labour to manage their farms [

Contrary to Cropping family labour, hired labour for redistribution units (LABHIRUC) had a positive effect on the direction of NPK nutrient balances. It significantly predicted phosphorus (P) balances with a unit change in LABHIRUC resulting in a change of 0.023 (p < 0.01) units in phosphorus balances.

Average slope of land, though a biophysical factor, was considered a proxy to land endowment resource quality. Farmers’ management practices and prices farmers are willing to offer for a given piece of land tend to differ depending on slope percentage, perceived degradation and ease of management attributed to slope effect. The study observed a negative correlation between average slope (%) of land and NPK nutrient balances and that average slope was significantly and negatively correlated with nitrogen balances.

The study observed mixed results with regards to effects of household resource capital on nutrient balances. While the effects of “total capital owned” significantly lowered NPK nutrient balances, livestock-related capital (value of livestock) had significant positive effect (

Based on a two-level multilevel (multivariate multilevel) model fitted to the nutrient balance dataset, this study has shown that farm nitrogen mining is taking place and is putting the quality of arable land in Kenya at stake. However, and contrary to on-going narratives on blanket existence of widespread nutrient mining in Kenya, evidences from this study indicate that farm phosphorus and potassium balances are not always negative.

Agro-climatic zones are characterised by different biophysical potentials that may influence farm nutrient balances to different degrees. The study draws the conclusion that farm nitrogen, phosphorus and potassium balances do differ between agro-climatic zones classified as arable land in Kenya. For example, variances for farm nitrogen and phosphorus were significantly different from zero across agro-climatic zones. The same was corroborated by likelihood ratio test. This serves to indicate the necessity of designating agro-climatic zone specific nutrient management interventions to address declining quality of arable land rather than the use of blanket intervention approaches.

Household resource endowments and resource flow and allocation patterns have a potential to influence farm nutrient balances. This study explored the effects of household resource endowments on nutrient balances in arable land. The study concludes that livestock household resource endowments is an important determinant of nutrient balances at smallholder farm level, thus recommends improvement of livestock practices at farm level not only to improve on farm nutrient balances but also to increase farm-profitability. However, it is further noted that ownership of large volumes of capital (total value of capital) and family labour resources do not automatically translate into positive effects on farm nutrient balances, but rather it is the type of capital owned (e.g. livestock) and what use it has been put to that matters.

The study generated interesting results and demonstrated that multivariate multilevel models can be used to conduct meta-analysis of farm nutrient balances and to explore arable land quality despite the small sample size. Future studies with large sample sizes and a large pool of relevant factors and covariates are, however, required to further give higher order insights beyond this study. This can be reinforced by meta-analysis that focuses on summary statistics and the use of simulation modeling to summarise inferences by random numbers rather than by point estimates and standard errors.

Sincere thanks to Dr. Edgar Otumba of Maseno University for his encouragement and to smallholder farmers who participated in research activities leading to generation of data used in this study.

Onduru, D.D. and Onyango, F. (2017) Applying Multivariate Multilevel Models to Explore Arable Land Quality in Sub-Saharan Africa: A Case Study in Kenya. Open Journal of Statistics, 7, 972-987. https://doi.org/10.4236/ojs.2017.76069