a BCCfe@sddlmZddlmZmZddlmZddlZddlZ ddl m Z m Z m Z ddlmZddlmZddlmZerdd lmZddlmZd d gZeGd d d ZeGdddZd#ddddddZddddd ZddZddZeGdddZd$ddd dd!d"d ZdS)%) annotations) dataclassfield) TYPE_CHECKINGN)special interpolatestats) CensoredData)ConfidenceInterval)norm)Literalecdflogrankc@seZdZUdZded<ded<eddZded<eddZded<eddZded <eddZ d ed <d d Z ddZ dddZ dddddZ ddZddZdS)EmpiricalDistributionFunctionaAn empirical distribution function produced by `scipy.stats.ecdf` Attributes ---------- quantiles : ndarray The unique values of the sample from which the `EmpiricalDistributionFunction` was estimated. probabilities : ndarray The point estimates of the cumulative distribution function (CDF) or its complement, the survival function (SF), corresponding with `quantiles`. np.ndarray quantiles probabilitiesF)repr_n_d_sfstr_kindc Cs||_||_||_||_|dkr$|nd||_||_|dkr@dnd}d|}t|dt|gtj tj g}t|dt|g||g} t j || ddd|_ dS)NsfrpreviousT)kindZ assume_sorted) rrrrrrnpinsertleninfrZinterp1d_f) selfqpndrZf0f1xyr*Q/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/stats/_survival.py__init__+s  z&EmpiricalDistributionFunction.__init__cCs ||S)zEvaluate the empirical CDF/SF function at the input. Parameters ---------- x : ndarray Argument to the CDF/SF Returns ------- y : ndarray The CDF/SF evaluated at the input )r!)r"r(r*r*r+evaluate<s z&EmpiricalDistributionFunction.evaluateNc Ksz ddl}Wn2ty>}zd}t||WYd}~n d}~00|dur\ddlm}|}ddi}||t|jd}|j} | d|gt | | d|g} |j | | | fi|S)a4Plot the empirical distribution function Available only if ``matplotlib`` is installed. Parameters ---------- ax : matplotlib.axes.Axes Axes object to draw the plot onto, otherwise uses the current Axes. **matplotlib_kwargs : dict, optional Keyword arguments passed directly to `matplotlib.axes.Axes.step`. Unless overridden, ``where='post'``. Returns ------- lines : list of `matplotlib.lines.Line2D` Objects representing the plotted data rNz2matplotlib must be installed to use method `plot`.wherepostg?) matplotlibModuleNotFoundErrorZmatplotlib.pyplotZpyplotZgcaupdaterZptprliststepr-) r"axZmatplotlib_kwargsr1excmessageZpltkwargsdeltar#r*r*r+plotKs    $z"EmpiricalDistributionFunction.plotffffff?linear)methodcCsd}|jdurt||j|jd}dt|d}||vrHt|d}t|d}|j svd|krtd ks~nt|||}||\}}d }t t |t |Brt j |td d t|dd t|dd }}t|j|dd|j}t|j|dd|j}t||S) a^Compute a confidence interval around the CDF/SF point estimate Parameters ---------- confidence_level : float, default: 0.95 Confidence level for the computed confidence interval method : str, {"linear", "log-log"} Method used to compute the confidence interval. Options are "linear" for the conventional Greenwood confidence interval (default) and "log-log" for the "exponential Greenwood", log-negative-log-transformed confidence interval. Returns ------- ci : ``ConfidenceInterval`` An object with attributes ``low`` and ``high``, instances of `~scipy.stats._result_classes.EmpiricalDistributionFunction` that represent the lower and upper bounds (respectively) of the confidence interval. Notes ----- Confidence intervals are computed according to the Greenwood formula (``method='linear'``) or the more recent "exponential Greenwood" formula (``method='log-log'``) as described in [1]_. The conventional Greenwood formula can result in lower confidence limits less than 0 and upper confidence limits greater than 1; these are clipped to the unit interval. NaNs may be produced by either method; these are features of the formulas. References ---------- .. [1] Sawyer, Stanley. "The Greenwood and Exponential Greenwood Confidence Intervals in Survival Analysis." https://www.math.wustl.edu/~sawyer/handouts/greenwood.pdf zKConfidence interval bounds do not implement a `confidence_interval` method.N)r=zlog-logz`method` must be one of .z4`confidence_level` must be a scalar between 0 and 1.r*rrzThe confidence interval is undefined at some observations. This is a feature of the mathematical formula used, not an error in its implementation.) stacklevel)rNotImplementedError _linear_ci _loglog_cisetlower ValueErrorrasarrayshapeanyisnanwarningswarnRuntimeWarningZcliprrrr )r"confidence_levelr>r8methodsZ method_funlowhighr*r*r+confidence_intervalqs4'      z1EmpiricalDistributionFunction.confidence_intervalc Cs|j|j|j}}}tjddd.|dt||||}Wdn1sX0Yt|}td|d}||}|j |} |j |} | | fS)Nignoredivideinvalidr@?) rrrrerrstatecumsumsqrtrndtrir) r"rOrr&r%varsezz_serQrRr*r*r+rCs<   z(EmpiricalDistributionFunction._linear_cic Cs |j|j|j}}}tjddd8dt|dt||||}Wdn1sb0Yt|}t d|d}tjdd"tt| }Wdn1s0Y||} t t ||  } t t ||  } |j dkrd| d| } } | | fS)NrTrUrr@rX)rVcdf) rrrrrYlogrZr[rr\expr) r"rOrr&r%r]r^r_Z lnl_pointsr`rQrRr*r*r+rDsF 0 z(EmpiricalDistributionFunction._loglog_ci)N)r<)__name__ __module__ __qualname____doc____annotations__rrrrrr,r-r;rSrCrDr*r*r*r+rs   &Hrc@s*eZdZUdZded<ded<ddZdS) ECDFResulta Result object returned by `scipy.stats.ecdf` Attributes ---------- cdf : `~scipy.stats._result_classes.EmpiricalDistributionFunction` An object representing the empirical cumulative distribution function. sf : `~scipy.stats._result_classes.EmpiricalDistributionFunction` An object representing the complement of the empirical cumulative distribution function. rrarcCs(t||||d|_t||||d|_dS)Nrar)rrar)r"r#rarr%r&r*r*r+r,szECDFResult.__init__N)rdrerfrgrhr,r*r*r*r+ris  risampleznpt.ArrayLike | CensoredDatarr )rj param_namereturnc Cs`t|ts\zt|d}WnBtyZ}z*t|d|}t|||WYd}~n d}~00|S)z.Attempt to convert `sample` to `CensoredData`.) uncensoredrmN) isinstancer rGrreplacetype)rjrker8r*r*r+_iv_CensoredDatas $rr)rjrlcCsft|}|dkr"t|}n&||jjkr>> sample = [6.23, 5.58, 7.06, 6.42, 5.20] # one-mile run times (minutes) The empirical distribution function, which approximates the distribution function of one-mile run times of the population from which the boys were sampled, is calculated as follows. >>> from scipy import stats >>> res = stats.ecdf(sample) >>> res.cdf.quantiles array([5.2 , 5.58, 6.23, 6.42, 7.06]) >>> res.cdf.probabilities array([0.2, 0.4, 0.6, 0.8, 1. ]) To plot the result as a step function: >>> import matplotlib.pyplot as plt >>> ax = plt.subplot() >>> res.cdf.plot(ax) >>> ax.set_xlabel('One-Mile Run Time (minutes)') >>> ax.set_ylabel('Empirical CDF') >>> plt.show() **Right-censored Data** As in the example from [1]_ page 91, the lives of ten car fanbelts were tested. Five tests concluded because the fanbelt being tested broke, but the remaining tests concluded for other reasons (e.g. the study ran out of funding, but the fanbelt was still functional). The mileage driven with the fanbelts were recorded as follows. >>> broken = [77, 47, 81, 56, 80] # in thousands of miles driven >>> unbroken = [62, 60, 43, 71, 37] Precise survival times of the fanbelts that were still functional at the end of the tests are unknown, but they are known to exceed the values recorded in ``unbroken``. Therefore, these observations are said to be "right-censored", and the data is represented using `scipy.stats.CensoredData`. >>> sample = stats.CensoredData(uncensored=broken, right=unbroken) The empirical survival function is calculated as follows. >>> res = stats.ecdf(sample) >>> res.sf.quantiles array([37., 43., 47., 56., 60., 62., 71., 77., 80., 81.]) >>> res.sf.probabilities array([1. , 1. , 0.875, 0.75 , 0.75 , 0.75 , 0.75 , 0.5 , 0.25 , 0. ]) To plot the result as a step function: >>> ax = plt.subplot() >>> res.cdf.plot(ax) >>> ax.set_xlabel('Fanbelt Survival Time (thousands of miles)') >>> ax.set_ylabel('Empirical SF') >>> plt.show() rz@Currently, only uncensored and right-censored data is supported.) rrZ num_censored_ecdf_uncensoredZ _uncensor_rightsize_ecdf_right_censoredrBri)rjresr8trarr%r&r*r*r+r s  cCsft|}tj|dd\}}t|}|j}||}d|}t|g||ddf}|||||fS)NT)Z return_countsrr0)rsortuniquerZru concatenate)rjr(countseventsr%rarat_riskr*r*r+rss  rscCs|j}|j}t||f}tdg|jdg|j}t|}||}||}t|jdd}tj|tj tj ddk}|dd}|dd} ||} ||} t || } tj| dd} t | | | }d|}| ||| | fS)Nrrr0)prependappend)r) _uncensoredrtrr{rHruZargsortZarangediffr rZZcumprod)rjZtodZtoltimesZdiedir~jZj_lZj_rrxr%cdr&rrar*r*r+rvs$    rvc@s"eZdZUdZded<ded<dS) LogRankResulta^Result object returned by `scipy.stats.logrank`. Attributes ---------- statistic : float ndarray The computed statistic (defined below). Its magnitude is the square root of the magnitude returned by most other logrank test implementations. pvalue : float ndarray The computed p-value of the test. r statisticpvalueN)rdrerfrgrhr*r*r*r+rs  r two-sidedz'Literal['two-sided', 'less', 'greater'])r(r) alternativerlcCs@t|dd}t|dd}tt|j|jft|j|jfd}t|}|jj t }|jj |}|jj |}|jj|}t|} t | jj |} t| jj d| } || } | | |||} |d|d}|dk} t| | || }|jj}t| ||}||t|}tj|t|}t|d|dd S) aCompare the survival distributions of two samples via the logrank test. Parameters ---------- x, y : array_like or CensoredData Samples to compare based on their empirical survival functions. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The null hypothesis is that the survival distributions of the two groups, say *X* and *Y*, are identical. The following alternative hypotheses [4]_ are available (default is 'two-sided'): * 'two-sided': the survival distributions of the two groups are not identical. * 'less': survival of group *X* is favored: the group *X* failure rate function is less than the group *Y* failure rate function at some times. * 'greater': survival of group *Y* is favored: the group *X* failure rate function is greater than the group *Y* failure rate function at some times. Returns ------- res : `~scipy.stats._result_classes.LogRankResult` An object containing attributes: statistic : float ndarray The computed statistic (defined below). Its magnitude is the square root of the magnitude returned by most other logrank test implementations. pvalue : float ndarray The computed p-value of the test. See Also -------- scipy.stats.ecdf Notes ----- The logrank test [1]_ compares the observed number of events to the expected number of events under the null hypothesis that the two samples were drawn from the same distribution. The statistic is .. math:: Z_i = \frac{\sum_{j=1}^J(O_{i,j}-E_{i,j})}{\sqrt{\sum_{j=1}^J V_{i,j}}} \rightarrow \mathcal{N}(0,1) where .. math:: E_{i,j} = O_j \frac{N_{i,j}}{N_j}, \qquad V_{i,j} = E_{i,j} \left(\frac{N_j-O_j}{N_j}\right) \left(\frac{N_j-N_{i,j}}{N_j-1}\right), :math:`i` denotes the group (i.e. it may assume values :math:`x` or :math:`y`, or it may be omitted to refer to the combined sample) :math:`j` denotes the time (at which an event occurred), :math:`N` is the number of subjects at risk just before an event occurred, and :math:`O` is the observed number of events at that time. The ``statistic`` :math:`Z_x` returned by `logrank` is the (signed) square root of the statistic returned by many other implementations. Under the null hypothesis, :math:`Z_x**2` is asymptotically distributed according to the chi-squared distribution with one degree of freedom. Consequently, :math:`Z_x` is asymptotically distributed according to the standard normal distribution. The advantage of using :math:`Z_x` is that the sign information (i.e. whether the observed number of events tends to be less than or greater than the number expected under the null hypothesis) is preserved, allowing `scipy.stats.logrank` to offer one-sided alternative hypotheses. References ---------- .. [1] Mantel N. "Evaluation of survival data and two new rank order statistics arising in its consideration." Cancer Chemotherapy Reports, 50(3):163-170, PMID: 5910392, 1966 .. [2] Bland, Altman, "The logrank test", BMJ, 328:1073, :doi:`10.1136/bmj.328.7447.1073`, 2004 .. [3] "Logrank test", Wikipedia, https://en.wikipedia.org/wiki/Logrank_test .. [4] Brown, Mark. "On the choice of variance for the log rank test." Biometrika 71.1 (1984): 65-74. .. [5] Klein, John P., and Melvin L. Moeschberger. Survival analysis: techniques for censored and truncated data. Vol. 1230. New York: Springer, 2003. Examples -------- Reference [2]_ compared the survival times of patients with two different types of recurrent malignant gliomas. The samples below record the time (number of weeks) for which each patient participated in the study. The `scipy.stats.CensoredData` class is used because the data is right-censored: the uncensored observations correspond with observed deaths whereas the censored observations correspond with the patient leaving the study for another reason. >>> from scipy import stats >>> x = stats.CensoredData( ... uncensored=[6, 13, 21, 30, 37, 38, 49, 50, ... 63, 79, 86, 98, 202, 219], ... right=[31, 47, 80, 82, 82, 149] ... ) >>> y = stats.CensoredData( ... uncensored=[10, 10, 12, 13, 14, 15, 16, 17, 18, 20, 24, 24, ... 25, 28,30, 33, 35, 37, 40, 40, 46, 48, 76, 81, ... 82, 91, 112, 181], ... right=[34, 40, 70] ... ) We can calculate and visualize the empirical survival functions of both groups as follows. >>> import numpy as np >>> import matplotlib.pyplot as plt >>> ax = plt.subplot() >>> ecdf_x = stats.ecdf(x) >>> ecdf_x.sf.plot(ax, label='Astrocytoma') >>> ecdf_y = stats.ecdf(y) >>> ecdf_y.sf.plot(ax, label='Glioblastoma') >>> ax.set_xlabel('Time to death (weeks)') >>> ax.set_ylabel('Empirical SF') >>> plt.legend() >>> plt.show() Visual inspection of the empirical survival functions suggests that the survival times tend to be different between the two groups. To formally assess whether the difference is significant at the 1% level, we use the logrank test. >>> res = stats.logrank(x=x, y=y) >>> res.statistic -2.73799... >>> res.pvalue 0.00618... The p-value is less than 1%, so we can consider the data to be evidence against the null hypothesis in favor of the alternative that there is a difference between the two survival functions. r()rjrkr))rmrightrr@rr*)rr)rrr rr{rrtr rrZastypeboolrrZ searchsortedrsumrur[rZ _stats_pyZ _get_pvaluer r)r(r)rZxyrwidxZtimes_xyZ at_risk_xyZ deaths_xyZres_xrZ at_risk_xZ at_risk_ynumZdenZsum_varZn_died_xZsum_exp_deaths_xrrr*r*r+rs2     )rj)r) __future__r dataclassesrrtypingrrLnumpyrZscipyrrrZscipy.stats._censored_datar Zscipy.stats._commonr Z scipy.statsr r Z numpy.typingZnpt__all__rrirrr rsrvrrr*r*r*r+s6       J "(