The psychometric function relates an observer's performance to an independent variable, typically a physical aspect of a stimulus. This paper, along with its companion, describes an integrated approach to fitting psychometric functions, assessing goodness of fit, and providing confidence intervals for parameters. The paper focuses on fitting and goodness-of-fit, while the companion paper addresses error estimation. The psychometric function models the relationship between stimulus intensity and performance, often using a sigmoid function like Weibull, logistic, or Gumbel. Fitting involves parameter estimation, error assessment, and goodness-of-fit testing. Maximum-likelihood estimation is used, with Bayesian priors to constrain parameters. A key issue is stimulus-independent errors (lapses), which can bias parameter estimates. The paper addresses this by allowing a free parameter for lapses. Simulations show that fixed lapse rates can lead to significant biases, while allowing lapses improves accuracy. The paper also discusses the use of Monte Carlo simulations instead of traditional chi-squared tests for psychophysical data, which often have small sample sizes. The results show that allowing lapses in parameter estimation reduces bias, especially for sampling schemes with high performance points. The paper concludes that varying the lapse parameter is essential for accurate threshold and slope estimation, despite potential inaccuracies in estimating the lapse rate itself. Goodness-of-fit assessment is also discussed, emphasizing the importance of using Monte Carlo simulations for small sample sizes, as asymptotic distributions may not be accurate. The paper advocates for these methods to ensure reliable parameter estimates and avoid biases in psychophysical data analysis.The psychometric function relates an observer's performance to an independent variable, typically a physical aspect of a stimulus. This paper, along with its companion, describes an integrated approach to fitting psychometric functions, assessing goodness of fit, and providing confidence intervals for parameters. The paper focuses on fitting and goodness-of-fit, while the companion paper addresses error estimation. The psychometric function models the relationship between stimulus intensity and performance, often using a sigmoid function like Weibull, logistic, or Gumbel. Fitting involves parameter estimation, error assessment, and goodness-of-fit testing. Maximum-likelihood estimation is used, with Bayesian priors to constrain parameters. A key issue is stimulus-independent errors (lapses), which can bias parameter estimates. The paper addresses this by allowing a free parameter for lapses. Simulations show that fixed lapse rates can lead to significant biases, while allowing lapses improves accuracy. The paper also discusses the use of Monte Carlo simulations instead of traditional chi-squared tests for psychophysical data, which often have small sample sizes. The results show that allowing lapses in parameter estimation reduces bias, especially for sampling schemes with high performance points. The paper concludes that varying the lapse parameter is essential for accurate threshold and slope estimation, despite potential inaccuracies in estimating the lapse rate itself. Goodness-of-fit assessment is also discussed, emphasizing the importance of using Monte Carlo simulations for small sample sizes, as asymptotic distributions may not be accurate. The paper advocates for these methods to ensure reliable parameter estimates and avoid biases in psychophysical data analysis.