The statistical analysis provides a quantitative foundation for interpreting the effects of plant growth–promoting bacteria (PGPB) on shoot length and grain yield,

Figure 2. Forest plot (Error bar) plot of shoot length differences across bacterial strains. This horizontal error bar plot displays the mean difference in shoot length (cm) for five plant growth–promoting bacterial strains compared to uninoculated controls. Each black dot represents the mean effect size, with horizontal lines indicating confidence intervals. All strains show positive mean differences, suggesting consistent vegetative growth enhancement. The dashed vertical line at 0 cm marks the null effect threshold, and none of the intervals cross it, reinforcing the statistical significance of the observed responses.

Figure 3. Forest plot of grain yield differences across bacterial treatments. This plot presents the mean difference in grain yield (kg/ha) for four individual bacterial strains compared to a baseline treatment of microbial consortium combined with half fertilizer input. Each black dot indicates the mean effect size, with horizontal error bars representing confidence intervals. All strains show positive yield differences, with varying degrees of precision, highlighting the comparative effectiveness of single-strain inoculations relative to the consortium benchmark.
integrating evidence across treatments and highlighting both consistency and variability in observed responses. The results presented in the forest plots and summary tables collectively demonstrate that PGPB inoculation exerts a measurable and positive influence on plant performance when compared with uninoculated controls. These outcomes are not isolated observations but emerge from weighted comparisons that account for variance and sample size, thereby strengthening the reliability of the conclusions drawn from the dataset.
Across all comparisons, the direction of effect sizes for shoot length is uniformly positive, indicating that PGPB treatments consistently enhanced vegetative growth relative to controls. Across all comparisons, shoot length responses were consistently positive, as illustrated in the forest plot (Figure 2). Individual bacterial strains differ in the magnitude of their effects, yet none show a negative mean difference. Shoot length outcomes following PGPB inoculation are summarized in Table 1. The pooled estimates reveal that treatments with narrower confidence intervals contributed greater statistical weight, reflecting higher precision in those estimates. Conversely, wider confidence intervals, as seen for some strains, indicate increased variability, likely arising from biological differences in plant–microbe compatibility or experimental conditions. Importantly, even in cases of broader intervals, the central tendency remains positive, suggesting that variability influences the strength rather than the direction of the response.The confidence intervals associated with most shoot length estimates do not cross the null value, providing statistical evidence that observed growth improvements are unlikely to be due to chance alone. This pattern underscores the robustness of the vegetative growth response to PGPB inoculation. The heterogeneity observed among strains, visualized by differing effect sizes and interval widths, aligns with the expectation that microbial traits such as phytohormone production, nutrient solubilization capacity, and rhizosphere colonization efficiency vary across taxa. Thus, the statistical dispersion captured in the forest plot reflects biological reality rather than analytical inconsistency.
Grain yield responses exhibit an even more pronounced statistical pattern. The forest plot for grain yield (Figure 3) shows substantially larger mean differences than those observed for shoot length, emphasizing that PGPB effects extend beyond early vegetative growth to influence final productivity. Grain yield responses to individual and consortium PGPB treatments are presented in Table 2. All treatments demonstrate statistically significant yield gains relative to controls, with confidence intervals that remain well above zero. The magnitude of these effects, indicates that PGPB-mediated yield enhancement is not marginal but agronomically meaningful.
Notably, the largest effect size is associated with the microbial consortium combined with reduced fertilizer input. This treatment not only exhibits the highest mean difference but also maintains a relatively narrow confidence interval, indicating both effectiveness and consistency. From a statistical perspective, this finding suggests a synergistic effect arising from multi-strain inoculation, where complementary functional traits collectively enhance nutrient acquisition and utilization. The weighting applied in the meta-analysis further reinforces this interpretation, as the consortium treatment contributes substantially to the pooled estimate due to its precision and effect magnitude.
The use of a random-effects model is particularly important for interpreting these results. By accounting for between-study and between-treatment variability, the model acknowledges that true effects may differ across experimental contexts. The persistence of statistically significant pooled effects under this conservative framework indicates that the positive influence of PGPB on both shoot length and grain yield is robust across heterogeneous conditions. Visual inspection of the forest plots (Figures 2 and 3) supports this conclusion, as the majority of individual estimates cluster on the positive side of the effect axis.
Potential publication bias for shoot length outcomes was evaluated using a funnel plot (Figure 4). Funnel plot analysis provides additional insight into the statistical reliability of the findings. The funnel plot for grain yield (Figure 5) displays a relatively symmetrical distribution of effect sizes around the pooled estimate, suggesting limited evidence of small-study effects or systematic bias. While the modest number of comparisons limits the power of formal bias detection, the absence of strong asymmetry supports the internal consistency of the dataset. This visual pattern indicates that large effects are not confined to small or imprecise studies, strengthening confidence in the observed yield responses.
Similarly, the shoot length funnel plot (Figure 4) shows an even spread of estimates across precision levels, with no

Figure 4. Funnel plot assessing publication bias in shoot length meta-analysis. This plot visualizes the relationship between standard error and effect size (mean difference in cm) for shoot length outcomes across studies. The vertical dotted line represents the pooled effect estimate, while the diagonal dashed lines define the expected distribution in the absence of bias. The symmetrical clustering of data points around the central line suggests minimal publication bias and supports the reliability of the aggregated results.

Figure 4. Funnel Plot Assessing Publication Bias in Meta-Analysis. This plot visualizes the relationship between study precision (standard error) and effect size (mean difference in kg/ha) across five studies. The red dotted line marks the overall mean effect, while the blue dashed funnel boundaries indicate expected variation in the absence of bias. Symmetry around the mean suggests low publication bias; asymmetry may indicate selective reporting or heterogeneity.
Table 1. Effect of PGPB Strains on Shoot Length. Comparison of shoot length (cm) in plants inoculated with different Plant Growth–Promoting Bacteria (PGPB) strains relative to uninoculated control plants. Standard deviations (SD) were calculated from reported standard errors (SE) using the formula: SD=SE×n, where n=3. Data can be used for forest plot analysis to compute Weighted Mean Differences (WMD).
|
Strain ID (PGPB)
|
Treatment Mean (cm)
|
Treatment SD
|
Treatment n
|
Control Mean (cm)
|
Control SD
|
Control n
|
|
RA-2 (A. faecalis)
|
27.9
|
5.47
|
3
|
15.9
|
3.86
|
3
|
|
RA-6 (P. arsenicoxydans)
|
28.2
|
5.97
|
3
|
15.9
|
3.86
|
3
|
|
RA-7 (B. aryabhattai)
|
28.9
|
6.15
|
3
|
15.9
|
3.86
|
3
|
|
RA-8 (P. brassicacearum)
|
25.7
|
6.88
|
3
|
15.9
|
3.86
|
3
|
|
RA-10 (P. azotoformans)
|
27.7
|
7.08
|
3
|
15.9
|
3.86
|
3
|
Table 2. Effect of PGPB Strains on Grain Yield. Mean differences in grain yield (kg/ha) relative to the control group. Standard errors (SE) are reported for funnel plot analysis to evaluate study precision and potential bias.
|
Strain / Treatment
|
Mean Difference (kg ha?¹)
|
Standard Error (SE)
|
Sample Size (n)
|
Significance (p-value)
|
|
RA-2
|
1219.39
|
249
|
3
|
< 0.05
|
|
RA-4
|
1258.55
|
293
|
3
|
< 0.05
|
|
RA-7
|
1306.33
|
393
|
3
|
< 0.05
|
|
RA-8
|
1202.79
|
416
|
3
|
< 0.05
|
|
Consortium + Half Fertilizer
|
2003.49
|
416
|
3
|
< 0.05
|
obvious clustering that would indicate selective reporting. The dispersion observed is consistent with expected biological variability rather than methodological distortion. Together, the funnel plots complement the forest plot findings by suggesting that the statistical outcomes are not driven by a subset of extreme or biased observations.
The numerical summaries provided in Tables 1 and 2 further contextualize these graphical results. Mean differences, confidence intervals, and weighting factors collectively illustrate how individual comparisons contribute to the overall conclusions. Treatments with higher variance exert less influence on pooled estimates, ensuring that results are not disproportionately shaped by uncertain data points. This weighting structure enhances the credibility of the pooled effects and aligns with best practices in meta-analytic methodology.
From an interpretive standpoint, the statistical results indicate a coherent progression from vegetative growth enhancement to yield improvement. Shoot length responses, while moderate in magnitude, signal improved early plant vigor, which often translates into greater photosynthetic capacity and resource acquisition. The substantially larger effect sizes observed for grain yield suggest that these early advantages persist and amplify throughout the growth cycle, culminating in higher productivity. The statistical separation between treatments, especially between single-strain inoculations and microbial consortia, highlights the added value of functional diversity in microbial applications.
In summary, the statistical analysis demonstrates that PGPB inoculation exerts consistent, positive, and statistically significant effects on both shoot length and grain yield. The convergence of evidence from forest plots, funnel plots, and summary tables supports the robustness of these findings despite inherent biological variability. By integrating effect size magnitude, precision, and model-based weighting, the results provide a strong quantitative basis for concluding that PGPB—particularly when applied as multi-strain consortia—represent effective biological tools for enhancing crop performance.
3.1 Interpretation and discussion of funnel and forest plots
The funnel and forest plots together provide a clear visual and statistical narrative of how plant growth–promoting bacteria (PGPB) influence shoot length and grain yield, offering insight not only into the magnitude of these effects but also into their consistency, variability, and reliability across treatments. Interpreted collectively, these plots strengthen the evidence base by allowing both effect estimation and assessment of potential bias within the analyzed dataset.
The forest plots illustrate the direction and size of PGPB effects relative to control treatments, with each horizontal line representing an individual treatment comparison and its associated confidence interval. For shoot length, the forest plot shows that all bacterial treatments are positioned on the positive side of the effect axis, indicating a uniform trend toward enhanced vegetative growth. Although the magnitude of response varies among strains, none demonstrate a negative effect. This pattern suggests that PGPB inoculation is inherently growth-promoting rather than conditionally beneficial. Differences in effect size are visually apparent, with some strains producing modest increases and others eliciting more pronounced shoot elongation. The variability in confidence interval width reflects differences in precision, which may arise from biological factors such as strain-specific metabolic activity or experimental variability in plant responses.
The pooled estimate for shoot length lies clearly to the right of the null line, indicating an overall statistically meaningful improvement when results are aggregated. The fact that this pooled effect remains positive despite the inclusion of strains with broader confidence intervals underscores the robustness of the vegetative growth response. In practical terms, this suggests that while the degree of benefit may vary, the probability of achieving growth enhancement through PGPB application is high across diverse bacterial taxa.
The forest plot for grain yield reveals an even stronger and more agronomically relevant pattern. Effect sizes for yield are substantially larger than those observed for shoot length, reflecting the cumulative impact of microbial inoculation over the full crop growth cycle. All treatments demonstrate positive yield responses, with confidence intervals well separated from the null value. This separation indicates that yield improvements are statistically reliable rather than artifacts of sampling variation. The visual prominence of the microbial consortium treatment highlights the advantage of combining multiple bacterial strains, as its effect size exceeds those of individual strains while maintaining relatively narrow confidence intervals. This combination of magnitude and precision suggests synergistic interactions that enhance nutrient availability and utilization.
The forest plots also convey important information about heterogeneity. The spread of effect sizes across treatments reflects biological diversity in microbial function and plant–microbe interactions. Rather than weakening the conclusions, this heterogeneity adds ecological realism, indicating that PGPB effects are modulated by strain composition and management context. The clustering of most estimates within a consistent positive range suggests that variability influences the strength of the response rather than its direction, reinforcing the reliability of the overall trend.
Funnel plots complement these findings by addressing the question of bias and study precision. In the shoot length funnel plot, the distribution of effect sizes around the pooled estimate appears relatively symmetrical. Studies with higher precision cluster near the top of the plot, while those with lower precision are more dispersed, forming an approximate inverted funnel shape. This pattern is consistent with expectations in the absence of strong publication bias. The lack of pronounced asymmetry suggests that small or imprecise studies do not disproportionately report exaggerated positive effects, supporting the internal consistency of the dataset.
Similarly, the grain yield funnel plot shows a balanced spread of effect sizes on either side of the pooled estimate. While some dispersion is evident at lower precision levels, there is no clear skew toward one side of the funnel. This symmetry indicates that large yield effects are not confined to a subset of small studies but are also present in more precise comparisons. Although the limited number of studies constrains the sensitivity of bias detection, the visual pattern does not raise concerns about selective reporting or systematic distortion of results.
When interpreted together, the forest and funnel plots reinforce one another. The forest plots demonstrate consistent positive effects with biologically plausible variability, while the funnel plots suggest that these effects are not driven by bias or methodological artifacts. The alignment between these graphical tools strengthens confidence in the meta-analytic conclusions, as both magnitude and reliability point toward the same inference: PGPB inoculation enhances plant growth and productivity.
Importantly, these plots also provide insight into the relative performance of single-strain versus consortium-based treatments. The forest plots show that while individual strains can significantly improve growth and yield, the most pronounced and stable effects are associated with multi-strain inoculation. The funnel plots support this interpretation by showing that consortium effects are not outliers but fall well within the expected distribution of effect sizes.
Overall, the visual evidence from the funnel and forest plots supports a coherent and statistically sound interpretation of PGPB efficacy. The consistency of positive effects, the manageable level of heterogeneity, and the absence of strong bias indicators collectively suggest that the observed benefits are genuine and reproducible. These plots therefore play a central role in validating the conclusion that PGPB, particularly in consortium form, represent reliable tools for enhancing crop growth and yield within sustainable agricultural systems.