Power analysis iPython notebook

power.ipynb

{
 "metadata": {
  "name": "Power Analysis"
 },
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     "source": [
      "# Power Analysis: Psychological Predictors of Poor ACLR Outcomes\n",
      "\n",
      "This analysis caclulates the power of detecting a non-zero effect of fear of movement/reinjury (TSK) or self efficacy (GSES) on recovery from ACLR. For Aim 1, we are estimating the power in predicting:\n",
      "\n",
      "- IKDC at year 2\n",
      "- KOOS quality of life at year 2\n",
      "\n",
      "For Aim 2, we are estimating power for predicting:\n",
      "\n",
      "- Single hop score\n",
      "- Crossover hop score\n",
      "\n",
      "For Aim 3, we are estimating power for predicting:\n",
      "\n",
      "- KOOS Sports and Rec\n",
      "- Marx score\n",
      "\n",
      "Power is estimated by generating simulated data from a model with hypothesized parameter values and standard errors, then using the simulated data to try to estimate the original model parameters. We are interested in determining whether the model can estimate the effects of the two risk factors with adequate precision. \n",
      "\n",
      "Model parameters for risk factors were estimated from a literature review, whereby extracted correlation coefficients ($\\rho$) were converted to linear model parameters ($\\beta$) via:\n",
      "\n",
      "$$\\beta = \\rho \\frac{\\sigma_y}{\\sigma_x}$$\n",
      "\n",
      "where $\\\\sigma_y$ and $sigma_x$ are the standard deviations of the response variable and the risk factor variable, respectively. Model parameters for covariates were estimated using pilot data (155 observations) to estimate linear models for each response variable. The estimated covariate parameters and risk factor parameters were combined in a single generative model to produce simulated data.\n",
      "\n",
      "Power was estimated by fitting Bayesian models to each of k simulated datasets for each response variable, and recording the proportion of calculated 95% Bayesian credible intervals for both risk factors excluded zero:\n",
      "\n",
      "$$ \\text{power} = \\frac{\\text{# intervals excluding zero}}{k} $$\n",
      "\n",
      "The implementation details of the power analysis are provided below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Correlations estimated from literature\n",
      "r_tsk_ikdc = r_tsk_qol = -0.22\n",
      "r_gses_ikdc = r_gses_qol = 0.25\n",
      "r_tsk_pcs = -0.3\n",
      "r_gses_pcs = 0.3\n",
      "r_tsk_mcs = -0.1\n",
      "r_gses_mcs = 0.2\n",
      "r_catas_ikdc = r_catas_qol = r_catas_pcs = r_catas_mcs = -0.2\n",
      "r_dep_ikdc = r_dep_qol = r_dep_pcs = -0.14\n",
      "r_dep_mcs = -0.3\n",
      "\n",
      "r_tsk_hop = -0.3\n",
      "r_gses_hop = 0.4\n",
      "r_catas_hop = r_dep_hop = -0.09\n",
      "\n",
      "r_tsk_sprec = r_tsk_marx = r_tsk_sport = -0.45\n",
      "r_gses_sprec = r_gses_marx = r_gses_sport = 0.6\n",
      "r_catas_sprec = r_catas_marx = r_catas_sport = -0.2\n",
      "r_dep_sprec = r_dep_marx = r_dep_sport = -0.14"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 90
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "'''\n",
      "Model-based covariate parameter estimates, along with correlations and standard errors. Each model includes:\n",
      "\n",
      "mu: parameter means\n",
      "se: parameter standard errors\n",
      "corr: correlation matrix\n",
      "sigma: standard deviation of response and observations, respectively\n",
      "r: correlations between risk factors and response\n",
      "\n",
      "'''\n",
      "\n",
      "# IKDC.raw.t2 ~ SEX + AGE + BMI.t0 + SMOKEc + KOOS.Pain.t2 + IKDC.raw.t0\n",
      "ikdc = {\n",
      "    'mu':[-1.584, 1.1385, 0.1513, -0.0341, -2.0032, 0.9421, 0.0158],\n",
      "    'se':[8.879, 1.195, 0.126, 0.173, 2.076, 0.075, 0.035],\n",
      "    'corr':[[1, 0.214, -0.086, -0.486, -0.033, -0.896, -0.193],\n",
      "        [0.214, 1, -0.035, -0.361, -0.158, -0.096, -0.193],\n",
      "        [-0.086, -0.035, 1, -0.407, -0.293, -0.015, 0.079],\n",
      "        [-0.486, -0.361, -0.407, 1, 0.128, 0.175, 0.123],\n",
      "        [-0.033, -0.158, -0.293, 0.128, 1, 0.065, -0.02],\n",
      "        [-0.896, -0.096, -0.015, 0.175, 0.065, 1, 0.028],\n",
      "        [-0.193, -0.193, 0.079, 0.123, -0.02, 0.028, 1]],\n",
      "    'sigma':[17,6.47],\n",
      "    'r':[r_tsk_ikdc, r_gses_ikdc]\n",
      "}\n",
      "\n",
      "# KOOS.KRQOL.t2 ~ SEX + AGE + BMI.t0 + SMOKEc + KOOS.Pain.t2 + KOOS.KRQOL.t0\n",
      "qol = {\n",
      "    'mu':[-54.96, -0.69, -0.33, 0.59, 1.51, 1.33, 0.08],\n",
      "    'se':[16.233, 2.231, 0.248, 0.323, 3.987, 0.144, 0.051],\n",
      "    'corr':[[1, 0.206, -0.118, -0.466, -0.097, -0.879, -0.159],\n",
      "        [0.206, 1, -0.007, -0.369, -0.139, -0.095, -0.104],\n",
      "        [-0.118, -0.007, 1, -0.387, -0.354, -0.015, 0.185],\n",
      "        [-0.466, -0.369, -0.387, 1, 0.194, 0.139, 0.109],\n",
      "        [-0.097, -0.139, -0.354, 0.194, 1, 0.113, 0.037],\n",
      "        [-0.879, -0.095, -0.015, 0.139, 0.113, 1, -0.087],\n",
      "        [-0.159, -0.104, 0.185, 0.109, 0.037, -0.087, 1]],\n",
      "    'sigma':[24,12.57],\n",
      "    'r':[r_tsk_qol, r_gses_qol]\n",
      "}\n",
      "\n",
      "# single_adj_geoavg_ratio ~ MTREAT1c + BMI.t0 + KOOS.Pain.t2 + MARX.t0\n",
      "hop = {\n",
      "    'mu':[1.0019, -0.0195, -0.0195, 0.0014, -0.0011, 0.004],\n",
      "    'se':[0.10299, 0.02151, 0.02525, 0.00177, 0.0009, 0.00155],\n",
      "    'corr':[[1, -0.172, -0.087, -0.544, -0.845, -0.193],\n",
      "        [-0.172, 1, 0.747, 0.052, -0.017, -0.078],\n",
      "        [-0.087, 0.747, 1, -0.08, -0.009, -0.136],\n",
      "        [-0.544, 0.052, -0.08, 1, 0.104, 0.231],\n",
      "        [-0.845, -0.017, -0.009, 0.104, 1, -0.101],\n",
      "        [-0.193, -0.078, -0.136, 0.231, -0.101, 1]],\n",
      "    'sigma':[0.18,0.07952],\n",
      "    'r':[r_tsk_hop, r_gses_hop]\n",
      "}\n",
      "\n",
      "# crossover_adj_geoavg_ratio ~ MTREAT1c + BMI.t0 + KOOS.Pain.t2 + MARX.t0\n",
      "hopc = {\n",
      "    'mu':[0.9274, -0.0117, -0.0497, 0.0021, -0.0002, 0.0041],\n",
      "    'se':[0.10646, 0.02226, 0.02612, 0.00184, 0.00093, 0.0016],\n",
      "    'corr':[[1, -0.173, -0.088, -0.545, -0.846, -0.192],\n",
      "        [-0.173, 1, 0.747, 0.052, -0.017, -0.077],\n",
      "        [-0.088, 0.747, 1, -0.08, -0.008, -0.136],\n",
      "        [-0.545, 0.052, -0.08, 1, 0.105, 0.231],\n",
      "        [-0.846, -0.017, -0.008, 0.105, 1, -0.102],\n",
      "        [-0.192, -0.077, -0.136, 0.231, -0.102, 1]],\n",
      "    'sigma':[0.09,0.08233],\n",
      "    'r':[r_tsk_hop, r_gses_hop]\n",
      "}\n",
      "    \n",
      "# KOOS.Sports_Rec.t2 ~ SEX + AGE + BMI.t0 + SMOKEc + KOOS.Pain.t2 + KOOS.Sports_Rec.t0\n",
      "sprec = {\n",
      "    'mu':[-16.916, 3.055, 0.024, -0.36, 3.611, 1.179, 0.038],\n",
      "    'se':[13.431, 1.868, 0.202, 0.266, 3.391, 0.117, 0.03],\n",
      "    'corr':[[1, 0.214, -0.086, -0.486, -0.033, -0.896, -0.193],\n",
      "        [0.214, 1, -0.035, -0.361, -0.158, -0.096, -0.193],\n",
      "        [-0.086, -0.035, 1, -0.407, -0.293, -0.015, 0.079],\n",
      "        [-0.486, -0.361, -0.407, 1, 0.128, 0.175, 0.123],\n",
      "        [-0.033, -0.158, -0.293, 0.128, 1, 0.065, -0.02],\n",
      "        [-0.896, -0.096, -0.015, 0.175, 0.065, 1, 0.028],\n",
      "        [-0.193, -0.193, 0.079, 0.123, -0.02, 0.028, 1]],\n",
      "    'sigma':[7.5,10.09],\n",
      "    'r':[r_tsk_sprec, r_gses_sprec]\n",
      "}\n",
      "    \n",
      "# MARX.t2 ~ SEX + AGE + BMI.t0 + SMOKEc + KOOS.Pain.t2 + MARX.t0\n",
      "marx = {\n",
      "    'mu':[6.727, 1.379, -0.141, -0.15, -0.462, 0.071, 0.258],\n",
      "    'se':[5.545, 0.738, 0.086, 0.107, 1.337, 0.048, 0.086],\n",
      "    'corr':[[1, 0.171, -0.161, -0.448, -0.087, -0.844, -0.248],\n",
      "        [0.171, 1, 0.037, -0.359, -0.134, -0.113, 0.059],\n",
      "        [-0.161, 0.037, 1, -0.381, -0.336, -0.036, 0.327],\n",
      "        [-0.448, -0.359, -0.381, 1, 0.191, 0.146, 0.038],\n",
      "        [-0.087, -0.134, -0.336, 0.191, 1, 0.108, 0.029],\n",
      "        [-0.844, -0.113, -0.036, 0.146, 0.108, 1, -0.097],\n",
      "        [-0.248, 0.059, 0.327, 0.038, 0.029, -0.097, 1]],\n",
      "    'sigma':[6,4.17],\n",
      "    'r':[r_tsk_marx, r_gses_marx]\n",
      "}\n",
      "\n",
      "models = {'ikdc':ikdc, 'qol':qol, 'hop':hop, 'hopc':hopc, 'sprec':sprec, 'marx':marx}\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 96
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The function `simulate_dataset` simulates data for a given model and sample size:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "\n",
      "def alphabeta(mu, sig):\n",
      "    # Calculates parameters of beta distribution from mean and sd\n",
      "    a = - mu * (sig**2 + mu**2 - mu) / sig**2\n",
      "    b = (sig**2 + mu**2 - mu) * (mu - 1) / sig**2\n",
      "    \n",
      "    return a,b\n",
      "\n",
      "def simulate_dataset(model, n=330, p_followup=0.85, rf_se=0.0001):\n",
      "    '''\n",
      "    Simulate a data for given model and sample size.\n",
      "    \n",
      "    '''\n",
      "    \n",
      "    # Fear of movement/re-injury\n",
      "    x_tsk0 = rnormal(43.3, 4.8**-2, n)\n",
      "    # y_tsk2 = rnormal(37.6, 6.8**-2, n)\n",
      "    \n",
      "    # Self-efficacy\n",
      "    x_gses = rnormal(30.64, 4.45**-2, n)\n",
      "    \n",
      "    # Covariates\n",
      "    x_age = rnormal(27, 11**-2, n).round(0)\n",
      "    x_male = pm.rbernoulli(0.56, n).astype(int)\n",
      "    x_smoker = pm.rbernoulli(0.22, n).astype(int)\n",
      "    x_bmi = rnormal(25.5, 4.5**-2, n)\n",
      "    x_pain_koos = pm.rtruncated_normal(75, 19**-2, 0, 100, n)\n",
      "    x_qol0 = rnormal(36, 20**-2, n)\n",
      "    mtreat = pm.rcategorical([1./3]*3, n)\n",
      "    x_mtreat_notrt = mtreat==1\n",
      "    x_mtreat_crepair = mtreat==2\n",
      "    x_marx0 = rnormal(16, 6**-1, n)\n",
      "    x_sprec0 = rnormal(45, 29.7**-1, n)\n",
      "    x_pain_cat = rnormal(14.4, 1.2**-2, n)\n",
      "    x_dep0 = pm.rtruncated_normal(12.5, 6.6**-2, 0, 27, n)\n",
      "    x_dep2 = pm.rtruncated_normal(7.2, 4.6**-2, 0, 27, n)\n",
      "    a,b = alphabeta(0.52,0.18)\n",
      "    x_ikdc0 = pm.rbeta(a,b,n)*100\n",
      "    \n",
      "    # Sample parameters\n",
      "    beta = pm.rmv_normal_cov(model['mu'], (np.array(model['corr'])*model['se']).T*model['se'])\n",
      "    \n",
      "    # Calculate betas for risk factors\n",
      "    beta_tsk = pm.rnormal(model['r'][0], rf_se**-2) * model['sigma'][0] / 4.8\n",
      "    beta_gses = pm.rnormal(model['r'][1], rf_se**-2) * model['sigma'][0] / 4.45\n",
      "    \n",
      "    # Build data matrix\n",
      "    if model == ikdc:\n",
      "        X = np.array([np.ones(n), x_male, x_age, x_bmi, x_smoker, x_pain_koos, x_ikdc0, x_tsk0, x_gses])\n",
      "    elif model == qol:\n",
      "        X = np.array([np.ones(n), x_male, x_age, x_bmi, x_smoker, x_pain_koos, x_qol0, x_tsk0, x_gses])\n",
      "    elif model == hop:\n",
      "        X = np.array([np.ones(n), x_mtreat_notrt, x_mtreat_crepair, x_bmi, x_pain_koos, x_marx0, x_tsk0, x_gses])\n",
      "    elif model == hopc:\n",
      "        X = np.array([np.ones(n), x_mtreat_notrt, x_mtreat_crepair, x_bmi, x_pain_koos, x_marx0, x_tsk0, x_gses])\n",
      "    elif model == sprec:\n",
      "        X = np.array([np.ones(n), x_male, x_age, x_bmi, x_smoker, x_pain_koos, x_sprec0, x_tsk0, x_gses])\n",
      "    elif model == marx:\n",
      "        X = np.array([np.ones(n), x_male, x_age, x_bmi, x_smoker, x_pain_koos, x_marx0, x_tsk0, x_gses])\n",
      "    else:\n",
      "        raise ValueError, \"Invalid model value\"\n",
      "    \n",
      "    # Build parameter array\n",
      "    B = np.r_[beta, beta_tsk, beta_gses]\n",
      "    \n",
      "    # Add noise\n",
      "    y = B.dot(X) + rnormal(0, model['sigma'][1]**-2, n)\n",
      "    # Random loss to followup, at rate p_followup\n",
      "    y[pm.rbernoulli(1-p_followup, n)] = -999\n",
      "    \n",
      "    return y, X"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 98
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The function `generate_model` specifies the model structure for given simulated data. Here, we use a simplified fixed effects model, with no group random effect, as these were estimated to be very small based on the pilot data, and there were only three sites in the dataset."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pymc as pm\n",
      "from pymc import rnormal\n",
      "\n",
      "def generate_model(y,X):\n",
      "    '''\n",
      "    Build a Bayesian model for estimating model parameters from simulated data.\n",
      "    '''\n",
      "    \n",
      "    # Model parameters\n",
      "    beta = pm.Normal(\"beta\", np.zeros(len(X)), np.ones(len(X))*0.001, value=np.zeros(len(X)))\n",
      "    # Extract risk factor parameters for later use\n",
      "    tsk = pm.Lambda(\"tsk\", lambda b=beta: b[-2])\n",
      "    gses = pm.Lambda(\"gses\", lambda b=beta: b[-1])\n",
      "    \n",
      "    # Calculate predicted values\n",
      "    mu = pm.Lambda(\"mu\", lambda b=beta: b.dot(X))\n",
      "    \n",
      "    # Residual variance\n",
      "    sigma = pm.Uniform(\"sigma\", 0, 100, value=10)\n",
      "    # Convert to precision scale\n",
      "    tau = sigma**-2\n",
      "    \n",
      "    # Create a masked for loss to followup\n",
      "    masked_values = np.ma.masked_equal(y, value=-999)\n",
      "    \n",
      "    # Likelihood\n",
      "    obs = pm.Normal(\"obs\", mu=mu, tau=tau, value=masked_values, observed=True)\n",
      "    \n",
      "    return locals()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 100
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, `calc_power` runs the power analysis, simulating `k` datasets, each of which are fit using Markov chain Monte Carlo (MCMC) with a given number of iterations."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calc_power(k=100, iterations=10000, burn=9000, verbose=0):\n",
      "    '''\n",
      "    Calculate power over k replicates for all models.\n",
      "    '''\n",
      "    \n",
      "    # Dict to hold risk factor nodes\n",
      "    nodes = {m:[] for m in models}\n",
      "    \n",
      "    # Loop over models\n",
      "    for model in models:\n",
      " \n",
      "        # Initialize number of times an effect is not detected\n",
      "        fail = np.zeros(2, dtype=float)\n",
      "    \n",
      "        # Loop over datasets\n",
      "        for i in range(k):\n",
      "            \n",
      "            if not i%20: print 'Completed dataset', i\n",
      "                \n",
      "            # Generate simulated dataset\n",
      "            y, X = simulate_dataset(models[model])\n",
      "        \n",
      "            # Initialize model\n",
      "            M = pm.MCMC(generate_model(y, X))\n",
      "        \n",
      "            # Run model\n",
      "            M.sample(iterations, burn, verbose=verbose)\n",
      "        \n",
      "            # Get intervals\n",
      "            intervals = M.beta.stats()['95% HPD interval'][-2:]\n",
      "        \n",
      "            # Check for failure to reject\n",
      "            fail += [_[0]<0<_[1] for _ in intervals]\n",
      "    \n",
      "            # Calculate power\n",
      "            power = (k-fail)/k\n",
      "            \n",
      "            # Append nodes for risk factors\n",
      "            nodes[model] += [[M.tsk, M.gses]]\n",
      "            \n",
      "        print 'Power for model %s is tsk=%f and gses=%f' % (model, power[0], power[1])\n",
      "        \n",
      "    return nodes"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "results = calc_power(100, 10000, 9000, verbose=-1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Completed dataset 0\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 20\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 40\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 60\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 80\n",
        "Power for model ikdc is tsk=1.000000 and gses=1.000000"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Completed dataset 0\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 20\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 40\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 60\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 80\n",
        "Power for model marx is tsk=1.000000 and gses=1.000000"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Completed dataset 0\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 20\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 40\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 60\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 80\n",
        "Power for model hop is tsk=0.660000 and gses=1.000000"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Completed dataset 0\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 20\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 40\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 60\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 80\n",
        "Power for model hopc is tsk=0.530000 and gses=1.000000"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Completed dataset 0\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 20\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 40\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 60\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 80\n",
        "Power for model qol is tsk=1.000000 and gses=1.000000"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Completed dataset 0\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 20\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 40\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 60\n",
        "Completed dataset"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 80\n",
        "Power for model sprec is tsk=1.000000 and gses=1.000000"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here are the 95% Bayesian credible intervals for IKDC, where we had 100% power:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pm.Matplot.summary_plot([_[0] for _ in results['ikdc']], \n",
      "    custom_labels=[''], xlab=\"Fear of Movement/Injury (TSK)\", x_range=[-1.2, 0.1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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A1dVV+tnV1RUlJSWVNpw1xe9//3u888478PLyQq9evar8vop7hjVo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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1ca804950>"
       ]
      }
     ],
     "prompt_number": 106
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pm.Matplot.summary_plot([_[1] for _ in results['ikdc']], \n",
      "    custom_labels=[''], xlab=\"Self Efficacy (GSES)\", x_range=[-0.1,1.4])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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JDgwoIpKefkuWtjWAMLyMxc1iiUh6jgZDTU0XAP5z2/z/3MZzXSphBUVE0jBT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\nA4qIiKTEgCIiIin9P7PuX7K9RlN+AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1ca8040d0>"
       ]
      }
     ],
     "prompt_number": 107
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here are the intervals for crossover hop, where power for TSK was not as high:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pm.Matplot.summary_plot([_[0] for _ in results['hopc']], custom_labels=[''], xlab=\"Fear of Movement/Injury (TSK)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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VSqnVh1yTDOzl3vEuxsWLSQBSFX28RGrACorsLjk52S4z+WytSLRXHdp3OrkzKfWDEcmP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\n6f8BYKD8NMrUwsgAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1d113e250>"
       ]
      }
     ],
     "prompt_number": 103
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pm.Matplot.summary_plot([_[1] for _ in results['hopc']], \n",
      "    custom_labels=[''], xlab=\"Self Efficacy (GSES)\", x_range=[-0.005,0.025])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1d177b7d0>"
       ]
      }
     ],
     "prompt_number": 108
    }
   ],
   "metadata": {}
  }
 ]
}