Motivated by the self-thinning meta-data, a random-effects meta-analysis model with unknown precision parameters is proposed with a truncated Poisson model for missing sample sizes. The random effects are assumed to follow a t distribution to accommodate outlying aggregate values in the response variable. The logarithm of the pseudo-marginal likelihood (LPML) is used for model comparison. In addition, in order to determine which thinning law is more supported by the meta-data, a measure called ``Plausibility Index (PI)" is proposed. A simulation study is conducted to examine empirical performance of the proposed methodology. Finally, the proposed model and the PI measure are applied to analyze the motivating self-thinning meta-data in details.