Several possibilities here, hard to give a clear answer without more details on sample prep etc, but here's a couple of suggestions.

How has your calibration curve come out? Does it go through the origin, i.e. zero concentration, zero absorbance? The negative absorbance may well come out along the calibration plot depending on how you've calibrated the instrument.

Some counter-ions can also interfere with the detector and give negative absorbance so it's important to make up your samples and standards as close as possible, e.g. matching oxidation state, same counter-ions. Some methods you can try are using hydrogen peroxide to oxidise all your metals and using the M+ oxides/hydroxides for standards, nitric acid to oxidise and use M+ nitrates for standards etc...

Some ideas to get you started anyway...

I agree with you Tom Dugmore. Is difficult to give a clear answer. Another possibility is that even in some practical experiment depending on the method of detector calibration and/or the calculation of Minium Level of Detection, the recording system of the detector will either give you a value just below the detecting level or record as zero.

Many factors can contribute to this and its really difficulty to narrow on one,ranging from digestion,dilutions,extraction if any to instrument calibration.

Quick fix to check out is that , ensure you have a good calibration curve probably with the best tested linearity (R 0.999). Its also proper to run laboratory quality control sample (will help observe variation, if any)

I mentioned in my paper it is due to impurities probably, as illustrated in the figures, after a while (elimination of impurities), Acid Yellow 17 (a dye) with negative removal efficiencies had positive removal efficiencies.

Article A comparative study on removal of four types of Acid Azo dye...