The specific criterion used to determine the order of fit was defined as follows: for the solute of interest, the order of the fit was progressively check details increased as long as the added osmotic virial coefficient increased Radj,RTO2 by at least 0.005. Another method of determining the order of fit for the osmotic virial equation is by using confidence

intervals calculated on the osmotic virial coefficients (and if applicable, the dissociation constant) at a given significance level. Specifically, when considering an increase in the order of fit, it should be verified that in the higher-order model, the confidence interval of the added coefficient does not include zero—if it does, then the higher-order model is not appropriate and, therefore, the

order of fit should not GSK J4 chemical structure be increased. It should be noted that this criterion is mathematically equivalent to conducting a t-test to evaluate the hypothesis that the regression coefficient that would be added (in the higher-order model) is equal to zero. For the i  th regression coefficient, βiβi a 100(1 − α  )% confidence interval can be calculated using [49] equation(29) βˆi±tα/2,n-pσβˆi,where σβˆi is the standard error of βˆi and tα/2,n-ptα/2,n-p is the right-tailed (α  /2)% point of the Student’s t  -distribution with n   − p   degrees of freedom. The standard error of βˆi is given by equation(30) σβˆi=σˆ2Sii,where SiiSii is the ii  th element of covariance matrix S̲=(F̲TF̲)-1, F   is the design matrix (see Appendix A), and σˆ2 is the estimated model variance, defined by equation(31) σˆ2=∑(y(a)-yˆ(a))2n-p.In this work, a criterion based on a 95% confidence interval (i.e. α = 0.05) was used. It should be noted

that for electrolyte solutes, which require a dissociation constant and thus use the forms of the osmotic virial equation in Eqs. (9) and (10), the regression coefficients do not equal the osmotic virial coefficients. As a consequence, the calculation of confidence intervals on the osmotic virial coefficients of electrolyte solutes requires the use of error propagation equations to obtain the corresponding standard errors (e.g. see Bevington and Robinson [4]). Once all required coefficients had been obtained, the three non-ideal models (i.e. the molality- and mole fraction-based multi-solute until osmotic virial equations and the freezing point summation model) along with the ideal dissociation model and the molality- and mole fraction-based ideal dilute models were used to predict osmolalities in several multi-solute solution systems of cryobiological interest for which experimental data [3], [14], [19], [21], [24], [52], [66], [75] and [78] were available in the literature. For the freezing point summation model (Eq. (21)), freezing point depression predictions were converted to osmolality predictions using Eq. (3). For both mole fraction-based models (Eqs. (17) and (19) and Eqs.