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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Principles of statistics in Statistical Parametric Mapping"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This notebook is a basic introduction to the statistics of statistical parametric mapping (SPM). The page is designed for\n",
      "readers who have had very little formal statistical background, and\n",
      "I have tried to keep formulae to a minimum. \n",
      "\n",
      "For a more technical treatment of the statistical background, see the [chapter on the general linear model](http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf)\n",
      "in the book [Human Brain Function (2nd edition)](http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we load the interactive graphics and other libraries:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I like gray images, with nearest neighbor resampling:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matplotlib.rcParams['image.cmap'] = 'gray'\n",
      "matplotlib.rcParams['image.interpolation'] = 'nearest'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Introduction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The statistics interface to functional imaging packages can be hard to understand for those of us trained to think of statistics in terms of means and variances.\n",
      "\n",
      "The reason that the statistical interface to functional imaging packages is so\n",
      "much harder to understand is that SPM, FSL and others choose to take the user much closer to\n",
      "the math; they sacrifice ease of understanding for great flexiblity of the\n",
      "designs that can be analysed with the same interface. \n",
      "\n",
      "To understand these interfaces, \n",
      "you will need some understanding of the underlying math. I hope to\n",
      "show that this need not be an insurmountable task, even for the relative\n",
      "statistical novice.\n",
      "\n",
      "To start, it is important to point out that the functional imaging analysis creates\n",
      "statistics by doing a seperate statistical analysis for each voxel. Standard analyses analyze each voxel independently.\n",
      "Specifically, the standard analysis:\n",
      "\n",
      "1. does a regression analysis __separately at each voxel__;\n",
      "2. makes t statistics from the results of this analysis, for each voxel;\n",
      "3. works out a p value for each t statistic;\n",
      "4. shows you an image of the t statistics;\n",
      "5. suggests a correction to the significance of the t statistics which takes account of the multiple comparisons in the\n",
      "   image.\n",
      "\n",
      "This document tries to explain how steps 1 to 4 work. I have written a\n",
      "separate tutorial on Random Field Theory that describes the basics of step 5.\n",
      "\n",
      "Before continuing, it is worth re-stressing that the regression analysis is entirely independent for each voxel.\n",
      "\n",
      "The statistics it uses to do this are fairly straightforward, and can be found in most statistics textbooks.\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Naming of parts"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This section goes through the standard statistical terms for an analysis, and\n",
      "how they relate to a functional imaging analysis.\n",
      "\n",
      "The __response variable__ in statistics is some measured data for each\n",
      "observation. A response variable is often referred to as __dependent__\n",
      "variable. In the case of functional imaging, the response variable is made up\n",
      "of all the values from an individual voxel for each of the scans in the\n",
      "analysis.\n",
      "\n",
      "The __predictor variable__ contains some value used to predict the data in the\n",
      "response variable. A predictor variable may also be called an __independent__\n",
      "variable. Each variable contains a possible __effect__. In our case a\n",
      "predictor variable might be some covariate such as task difficulty which is\n",
      "known for each scan, and which might influence the response variable - in our\n",
      "case, voxel values.\n",
      "\n",
      "Thus, in functional imaging:\n",
      "\n",
      "* *observation* : a voxel value, in the voxel we are analysing, for one scan;\n",
      "* *response variable* : data for all the scans for one voxel (i.e. all the observations);\n",
      "* *predictor variable* : covariate = effect.\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Getting the data"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In order to run this tutorial, you'll need to \n",
      "[download](http://imaging.mrc-cbu.cam.ac.uk/downloads/Tutscans/egscans.tar.gz) \n",
      "and unpack a set of data files.\n",
      "\n",
      "The following two commands will download and unpack them on linux:\n",
      "\n",
      "    curl -O http://imaging.mrc-cbu.cam.ac.uk/downloads/Tutscans/egscans.tar.gz\n",
      "    tar xzf egscans.tar.gz\n",
      "\n",
      "The `fetch_scans()` function below will do all the downloading\n",
      "in a cross-platform manner, using python-only tools.  You should run it even if you downloaded the files\n",
      "manually, as it will compute the necessary paths for the rest of the script (it won't re-download files\n",
      "already present)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import os\n",
      "import tarfile\n",
      "import urllib\n",
      "\n",
      "def fetch_scans():\n",
      "    # Names of the files in the data bundle\n",
      "    scans_dir = 'egscans'\n",
      "    files = [os.path.join(scans_dir, 'snn03055dy%i.img' % i) for i in range(1,13)]\n",
      "    \n",
      "    # The remote url and filename for local storage\n",
      "    data_url = 'http://imaging.mrc-cbu.cam.ac.uk/downloads/Tutscans/egscans.tar.gz'\n",
      "    tarball = 'egscans.tar.gz'\n",
      "\n",
      "    # First, check that we don't already have the files, whose names are:\n",
      "    if all(map(os.path.isfile, files)):\n",
      "        print 'All files already present, nothing to download.'\n",
      "        return files\n",
      "    \n",
      "    # If we don't have them, make the storage directory, fetch (if needed)\n",
      "    # and unpack skipping the .mat files that are in the tarball.\n",
      "    if not os.path.isdir(scans_dir):\n",
      "        os.mkdir(scans_dir)\n",
      "\n",
      "    if not os.path.isfile(tarball):\n",
      "        print \"Fetching remote tarball, this may take some time depending on your network...\"\n",
      "        urllib.urlretrieve(data_url, tarball)\n",
      "\n",
      "    print \"Unpacking scans, omitting .mat files\"\n",
      "    tf = tarfile.open(tarball, 'r:gz')\n",
      "    image_files = [ f for f in tf.getmembers() if not f.name.endswith('.mat')]\n",
      "    tf.extractall(scans_dir, image_files)\n",
      "    print \"All files unpacked into\", scans_dir, ', done!'\n",
      "    return files\n",
      "\n",
      "files = fetch_scans()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "All files already present, nothing to download.\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Starting the analysis"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's now make actual images out of these files"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import nibabel as nib\n",
      "images = map(nib.load, files)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# grand mean for data\n",
      "GM = 50"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we load the affine transformation from the first image (it's the same for all) and extract the conversion\n",
      "parameters from voxels to MNI space (in mm)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# transformation from voxel no to mm\n",
      "M = images[0].get_affine()\n",
      "# mm to voxel no\n",
      "iM = np.linalg.inv(M)\n",
      "# coordinates of voxel of interest in mm (MNI space)\n",
      "posmm = [-20.0, -42, 34]\n",
      "# coordinates in voxel space (in homogenous coordinates)\n",
      "posvox = np.dot(iM, posmm + [1])\n",
      "# We grab the spatial part of the output.  Since we want to use it as an \n",
      "# index, we need to make it a tuple\n",
      "posvox = tuple(posvox[:3].astype(int))\n",
      "posvox"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "(29, 35, 42)"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We need to define a small utility function commonly used in SPM, we'll call it `global_mean`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def global_mean(img):\n",
      "    \"\"\"Compute a \"sorted global mean\" of an image.\n",
      "\n",
      "    This returns the mean of the image voxels that are in the range above \n",
      "    1/8 of the mean.\"\"\"\n",
      "    \n",
      "    m = img.mean()/8.0\n",
      "    return img[img>m].mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With this utility, we can now get the data for the voxel of interest across all the images, and compute the global mean of each scans:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "nimgs = len(images)\n",
      "vdata = np.zeros(nimgs)\n",
      "gdata = np.zeros(nimgs)\n",
      "for i, V in enumerate(images):\n",
      "    d = V.get_data()\n",
      "    vdata[i] = d[posvox]\n",
      "    gdata[i] = global_mean(d)\n",
      "gdata"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "array([ 0.00459615,  0.004154  ,  0.00462038,  0.00439584,  0.00417357,\n",
        "        0.00442748,  0.00416478,  0.00471157,  0.00419319,  0.00455416,\n",
        "        0.00420025,  0.00405441])"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can display slices from the scans.  Here is slice 30 of the last scan:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "last_image = images[-1]\n",
      "last_data = last_image.get_data()\n",
      "imshow(last_data[:, :, 30])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "<matplotlib.image.AxesImage at 0x717f350>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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NnReMe9qek36c3JOSmoXELQ8lBXZyY3DvRykqXGvGoz4Dbs6S7NEkD9R896R5\nSmci496jcLxifcXGprZbwu9m+P2VtjGOHj2KY8eOAQCGh4fF+YsL1XvvvYeLLroI//vf/9DY2Ii6\nuroJP0+lUuwDV1ZW5tvRIuU4jrN06VIsXboUADAwMIDe3t6i14t/3i666CIAwLJly3DXXXdh3759\nSKfT6OvrAwD09vaiqqpqqvN2HMdhKWpRffXVVxgbG8OSJUtw8uRJvPnmm3jiiSfQ1NSETCaDzZs3\nI5PJoLm5WbwRJSc4WcfJHsqE1KSZ4GSIlLdcIwmpcTm4IDcqN7emLUk/jSyi5BwnEy1eVU2wqZTA\nL4QLTixFhW3ue0NVWy6FxONKZ3HvE5XgkTtbKOW753Ldh3DPKKVWsrQ5zyFH0YUqm83irrvuyg98\n33334dZbb8XatWuxYcMGbN++PR+e4DiOUy6KLlQXX3wx9u/fP6m/srISHR0d4uBScjVuU47boKP6\nrelqqbbmCIIUJ6M53mBJq6yJnZJS3kpxOQD9V9yaXpYaQ+PMCKGsXs4S1sQQUX2aOYVtKjOGdeOa\nehbN0RvJouLiuc466yyyP7qey56gib+jnD5h+9SpU2Sbut6SrhnwyHTHcWYBvlA5jpN4ypo9QTr+\nwskbjWkZvZaTRZrjF5T0C9H0W4plSknvuA1GzZEFaq7ccYq4MpYb2yJ1NDnOqfzemvEsm+mcdJU2\ndjUSVKqGwyVk5D5PysnByT2uHV2vye0vZUcIfyfDGKiwzclAKklkSY7QOI7jzDS+UDmOk3imLXEe\nJXs4Tx9nNlJtTiaGZi/n4aHMYGtebUvGAUsZImvO9GgeGo8Yl2ObwiJvwrYm2Zx0zMbq6bPkhecy\nSFDjce+dJn4p+l5z12qkn8Xrd/bZZ5P90Wu5Z+GkH/V7yck9jQyUkkRyuEXlOE7i8YXKcZzEU1bp\nF7fKMefpo0zLRYsW5ftCszccjyshZCmppDmSQ12rkYTU+2GtRhvBeeksWQs0koZ7T6nAQk76SR47\nzediyZ4QYvFmcnJVI4Up2abJgU/NifP6cW1qTtwWBif3qN/FoaGhfJ+mLW3ZuPRzHGdO4AuV4ziJ\nZ0YqJVNBX5z3jvMeRP3hz0PpF7ZDDyAV4KhJemepHst5lLjxNOfHtGgCICUvHCdjNOfLJOlnOaen\ngXtdXA8gJY2sco8qnaaR0FK5N430lnL3cwHSnNc9/P366quvAEyUdVFfYT/nAaTO+rn0cxxnTuAL\nleM4iWct3K86AAAQD0lEQVRGpJ+UOI8L8gxNy0jaaeQeFfgWtqfi1YmeRXOWjhujFGfsqD4piRog\ne+w4uSe1uffc4n0stTzmUr5I/ZynjDsjSMlbLkGe5Xyq5vPkZCyVD5/bYuGkXdQ+efIk+fOw3wM+\nHcc545g2i0pa3TXWFbXqc6fFNZuN1LECDikTgTW5mpQojkuoJhW35DarLZYRd63F0tJYVFJSO01C\nQ4ulpYnLosbjjthoEr5RSNY5d73m+FEIFRslbZQDEy2jwcHBSf1hX/g6P0LjOM4ZjS9UjuMkHlH6\nDQwM4Cc/+Qk+/fRTpFIpvPDCC1i1ahU2btyIQ4cO5Ys7VFRUTHqtFK6vKWQYmpDUBjknYziZIpnM\n3BEEDsosD8fgpBi1KavJ0W6RfmGbez8o6cxJaM17TclHTeWTEGprgJNcpZCBUr81t7j0jJqsENT1\n3OssmQ9CqRZKuBMnTpDtL7/8ctL13FEZKXYqbJc8cd6jjz6K22+/HQcOHMDHH3+Muro6tLW1obGx\nEV1dXWhoaEBbW5t4I8dxnLgUXaiOHz+Od999Fw8++CCA038Zzz//fLS3t6OlpQUA0NLSgl27dpV/\npo7jnLEUlX4HDx7EsmXL8MADD+Dvf/87rrvuOjzzzDPIZrNIp9MAgHQ6jWw2S75eKuDJJYoLTcVQ\nNoRmZtxT+hShucwlHpPM+fDeIZz3K7xndI21bFcIlThPE1NFHYXRSDwppkcT58NhOf6ieW8s95AK\n1GpkGxWLpZF4FumqyXxAxSCGUo6SdZprQq8glyWB28qhfvenLP2+/vprdHZ24pFHHkFnZycWL148\nSealUin2Q//yyy/z/0LN6jjOmc3IyAgGBwcxODg4YbHjKLpQ1dTUoKamBtdffz0A4J577kFnZyeq\nq6vR19cHAOjt7UVVVRX5+iVLluT/hXmjHMc5sznrrLNw3nnn4bzzzsM555wjXl9U+lVXV2PFihXo\n6upCbW0tOjo6sHr1aqxevRqZTAabN29GJpNBc3Mz+XpL4jwuiV5oTkrBmtb82ZGpzZmhnPeLkpWc\nzKLuV4gkWazlvArnVmx+Fo+dRs5JHiqN54oKgA3RJAeM+95Q10g51YuNERfKm8klYeRkVmipRHIt\nlHLHjx8n25LXjwvylEpkhfPWVDgPEcMTnnvuOdx3330YGRnBJZdcghdeeAFjY2PYsGEDtm/fng9P\ncBzHKRfiQnX11VfjL3/5y6T+jo6OskzIcRynkGk762eplKzJgy0F5nHzoObESVAuMwPlybMmh5Ny\nhJcabk6UnLMmZSu1BJLGkDxz3Bhx5xn3jGDYbw3mpbIqaGoMcJkPIgmnkX5hmzrXx0k/7vyeVBpO\nI/38CI3jOInHFyrHcRLPjJTLisxCa9qKuEnXOJOayt0emq/nnnvupGsBWgZq0ppIifikysGF/XGR\nqh9z3r24Hi/NebxSn9mTrrXIubh59MO29B0sdk30ndTUFQilHxXEyZ3pkzx9wDeeQ867x5XAoqSf\nVPatELeoHMdJPNNmUVHZAjQJszRjR2isKEshVE1/FMgajhvGX2msxqhfE7NUzo1rKn6Ji7+SrAzO\nkaKBuj5uxoSpxHNJ95MSQwKyw0aTMFJKekfFSwG0RcUdj+GsKKrKDHdUxlJA12pBu0XlOE7i8YXK\ncZzEU1bpJ1XmmI5jB4VtqRCqpi3FhWhieyjHgCXWqfD6UmKNkZKOe3AJA6UxNNkkOEoRZyfNg/tO\nUFsGoUTi2uFmNCXzuIR1XDI8qs39nKssQyXG4zbTueKmknPBpZ/jOHMCX6gcx0k8MxpHVU4sHkDu\n55a2paQVQMs8qiip5nWF7elAOpbEvf9x5SN3bYh0LIkrOsqNLR314pI9hm0pBorz5FFePaoYKCCX\ntwrbVF/heKHco+bKZUZwr5/jOGc0vlA5jpN4ZiTgMzK7Oa+g5giN5diGdLxBY7JyXh1J+pWi1JXm\nuUuRgcFydCWE+hzjyr2wbZUHkvTTBHlK2TU0HjvqeItG7nGePCprASfhODkX9WtKXXHHYqhSVxa5\nB8T/bN2ichwn8fhC5ThO4pm2xHmUSR2a3xoZGJqZlhJYUvAnd610RjDstwaKWoLgOBkbZmyI+qfb\n+zcV4kq/uF4/DVJAsEbuUQGRceUeQHvsQu+eZozonpZqxgDt5dQEukrfdes2w+z5VjuOc8ZSdKH6\n7LPPsGbNmvy/888/H88++yz6+/vR2NiI2tpa3HrrrRgYGJiu+TqOcwaSyiltsPHxcSxfvhz79u3D\nc889hwsvvBC//vWvsXXrVhw7dowsTBrW66LOrknVdgE+fzdVKVnTpioDh4nwwvqDYTtMoke1pZ9b\nxwjfu3AeYQoZqoqxJgjUmvd7qljPYFIBn5YqyFOZH+W5snr6KOnHyT0uRQsl4TjvnsaTR53T0zwX\nJYWtQdHUtknh5z0yMlL0+6eWfh0dHbj00kuxYsUKtLe3o6WlBQDQ0tKCXbt2aYdxHMcxo95M37Fj\nB+69914AQDabRTqdBgCk02lks1nTTS2bpVIsFjVuYVta6TVxT1IqWevp+LjHLELrKrSooramUsx0\no/lsJedC3MR5Gkss7mfLfUZUv9WiojbIuTgqTRHQqK05DiSlSrY4mQrbZY2jGhkZwauvvoof/vCH\nk36WSqXYX4LR0dH8P2uWR8dx5i7j4+MYGxvL/5NQLVRvvPEGrrvuOixbtgzAaSuqr68PANDb24uq\nqirydQsXLsz/48qcO45z5jFv3jzMnz8//09CJf1efvnlvOwDgKamJmQyGWzevBmZTAbNzc3xZ/z/\nxJWBmmvDzXTKzORkhUUGcia1pmJH1B9upofmvGVjnXNKhJS6eKiUczyutLYeRZLy6Gu2FOJKP0m2\nc3FUXDFPSvppNs25eUhJIi1yzvp5SnFUJZF+J0+eREdHB+6+++58X2trK/bs2YPa2lrs3bsXra2t\n4o0cx3HiIlpUixcvxpEjRyb0VVZWoqOjo2yTchzHCSnrEZqQuAUrJa8NZ0KG8sYSr8N5C7l+izzg\nJCGVE5uTe5zXL4oFC2WupVRXYX+xvmL9EXEldNjWxOVIMtA6D0oaaTy6XDK5qJ8rGKo5WiMdf9HM\ng9qisHrsLO+p5XdOgx+hcRwn8fhC5ThO4imr9CtFkKFkQnI/DyVNXC8GJyslk1oTFBge24nM/NCL\nF0q8UAaGrwuvjyQfdawG0MlAS8YBKUsDZ9pzEqPUMsUi6yXpp0mcJ0l/y3Eb7hpO7nFtS27/uLJN\nU9JM+r3V4BaV4ziJxxcqx3ESz7RJP4vE4MxCqtwRVwKJk4GUN3AqZw6pxHmcmU9lbgC+CegLZR0n\n8bh2NHYo9zQVlikZqPH0SV5c6/soST9rTnopGaEl4JMLkuTOylFSbCpnQalc5Ra5B8Tf5pDKmFle\nF7Zd+jmOM+fwhcpxnMQzbQGfFBrzz2IiaspNSdJPkwgslFeROc4l6gs9OZxHLmpzSfEsbU0ywhBK\nEpZC+oXElVyWRGyF95G8wpZgXs35OE5+RddT5+607WhsSyoW7hnjSrVSjxHi5bIcx5kTzIhFJa2g\nFktL89dfiq8Kfx5aGJoNYcqi4iyZ8N7U9da0ylRbkziP21i3FO3kiO7JxVFJ1kv4Wi6+yZLO2Brn\nQ1kkobUkxX5xz8VZS5bxrGl/LdZQiNQf13LS3IPDLSrHcRKPL1SO4ySeGSlAKl07lWsiOMlCyUDr\n6X5KOnFySiMDo7YkDYuNEfVTfYX9caVd3IR71pzplNTRvC7uvaWjNZw8szgGLFkjuGusiQRLsYkt\nva4UMlBDWS2qMyFPenj+ai5y9OjRmZ5C2Tlx4sRMT6GszIXfw7IuVNacM7ORub5Q9ff3z/QUys5c\nX6jmwu9h2aUfFXZvOaVv9UwU3qMQyuunybTAxWKNjY1hZGRElYyOyzgQSTSNZ07q18SPcUTXhGMM\nDQ3lF6u4WRWsHkCLx477Hlgk4cjICL788stJr5OOnWiO9VhkW9yEdRqJF3exiisZLeNp8M10x3ES\nj7qku3ngGSx46TjO7KPYUlQ26Vem9c9xnDMQl36O4yQeX6gcx0k8ZVuodu/ejbq6OqxatQpbt24t\n122mlZ6eHnzve9/D6tWrccUVV+DZZ58FcNqF39jYiNraWtx6660YGBiY4ZlOjbGxMaxZswbr1q0D\nMPeeb2BgAPfccw8uv/xy1NfX48MPP5xTz7hlyxasXr0aV155JX784x/j1KlTs/75yrJQjY2N4Re/\n+AV2796Nf/zjH3j55Zdx4MCBctxqWlm4cCGefvppfPrpp/jggw/w/PPP48CBA2hra0NjYyO6urrQ\n0NCAtra2mZ7qlNi2bRvq6+vzDpG59nyPPvoobr/9dhw4cAAff/wx6urq5swzdnd34/e//z06Ozvx\nySefYGxsDDt27Jj9z5crA++//37utttuy///li1bclu2bCnHrWaUO++8M7dnz57cZZddluvr68vl\ncrlcb29v7rLLLpvhmcWnp6cn19DQkNu7d2/ujjvuyOVyuTn1fAMDA7mLL754Uv9cecajR4/mamtr\nc/39/bnR0dHcHXfckXvzzTdn/fOVxaI6fPgwVqxYkf//mpoaHD58uBy3mjG6u7vx0Ucf4YYbbkA2\nm0U6nQYApNNpZLPZGZ5dfB5//HE89dRTEwI359LzHTx4EMuWLcMDDzyAa6+9Fg8//DBOnjw5Z56x\nsrISv/rVr/Cd73wH3/72t1FRUYHGxsZZ/3xlWajmegzV4OAg1q9fj23btmHJkiUTfpZKpWbt87/2\n2muoqqrCmjVr2PCS2fx8wOlDwp2dnXjkkUfQ2dmJxYsXT5JBs/kZv/jiCzzzzDPo7u7Gf/7zHwwO\nDuKll16acM1sfL6yLFTLly9HT09P/v97enpQU1NTjltNO6Ojo1i/fj02bdqE5uZmAKf/QvX19QEA\nent7UVVVNZNTjM3777+P9vZ2XHzxxbj33nuxd+9ebNq0ac48H3Dauq+pqcH1118PALjnnnvQ2dmJ\n6urqOfGMf/3rX3HjjTfiggsuwIIFC3D33Xfjz3/+86x/vrIsVGvXrsXnn3+O7u5ujIyM4JVXXkFT\nU1M5bjWt5HI5PPTQQ6ivr8djjz2W729qakImkwEAZDKZ/AI223jyySfR09ODgwcPYseOHbjlllvw\n4osvzpnnA4Dq6mqsWLECXV1dAICOjg6sXr0a69atmxPPWFdXhw8++ABDQ0PI5XLo6OhAfX397H++\ncm1+vf7667na2trcJZdcknvyySfLdZtp5d13382lUqnc1Vdfnbvmmmty11xzTe6NN97IHT16NNfQ\n0JBbtWpVrrGxMXfs2LGZnuqUefvtt3Pr1q3L5XK5Ofd8+/fvz61duzZ31VVX5e66667cwMDAnHrG\nrVu35urr63NXXHFF7v7778+NjIzM+ucr21k/x3GcUuGR6Y7jJB5fqBzHSTy+UDmOk3h8oXIcJ/H4\nQuU4TuLxhcpxnMTjC5XjOInn/wC24mmp1gl60AAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x45ee510>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Plot globals"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(gdata, vdata, 'x')\n",
      "plt.xlabel('TMVV for scan')\n",
      "plt.ylabel('Voxel value for -20 -42 34')\n",
      "plt.title('The relationship of overall signal to values for an individual voxel')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "<matplotlib.text.Text at 0x71a2b90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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8f+rUKSxevBi7d+/WW79hwmKMMdZU4x/zy5cvt1wwehjsEnz//ffRr18/pKSk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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x717f7d0>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# proportionally scale data\n",
      "pvdata = vdata / gdata\n",
      "# scale to global mean of 50\n",
      "Y = pvdata * GM\n",
      "# our covariate\n",
      "ourcov = np.array([5,4,4,2,3,1,6,3,1,6,5,2])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# plot data \n",
      "plot(ourcov, Y, 'x')\n",
      "xlabel('Task difficulty')\n",
      "ylabel('PS voxel value')\n",
      "xlim(0, 7)\n",
      "title('The relationship of task difficulty to voxel value');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MwLlz5wAAOp0ODg4OSElJqVB2z549mDZtGoqLixEeHo4ZM2Y80IkREdHfHnpC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\nj4944oknxKpVq4QQQkRHR4t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       "text": [
        "<matplotlib.figure.Figure at 0x7402150>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# guess intercept, slope, estimated points\n",
      "guessic   = 55\n",
      "guesslope = 0.5\n",
      "guessPts  = guessic + guesslope*ourcov"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For plotting it will be convenient to have the boundaries of our plot region available"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "minx = 0\n",
      "maxx = ourcov.max()+1 # min max x for plots\n",
      "axs = np.array([minx, maxx, Y.min()*0.98, Y.max()*1.02])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# plot data with guess intercept and slope\n",
      "plot(ourcov, Y, 'x')\n",
      "plot(axs[:2], guessic+guesslope*axs[:2], ':') # guessed line\n",
      "# plot guessed residuals\n",
      "for i, c in enumerate(ourcov):\n",
      "  plot([c, c], [Y[i], guessPts[i]], color='r')\n",
      "plt.axis(axs)\n",
      "xlabel('Task difficulty')\n",
      "ylabel('PS voxel value')\n",
      "xlim(0, 7)\n",
      "title('A first guess at a linear relationship, and its residuals');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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WX375Jf7zn/8gPz+/KWIjhDSR5UeW48o/T8e7erndyzAT0XjxzUmdNfvTp0+j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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x769c910>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# residuals from guess slope\n",
      "guessRes = Y - guessPts"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we construct the design matrix $X$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "X  = np.column_stack([ourcov, ones(nimgs)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We would like to display it in a useful way"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def scale_design_mtx(X):\n",
      "    \"\"\"utility to scale the design matrix for display\n",
      "\n",
      "    This scales the columns to their own range so we can see the variations \n",
      "    across the column for all the columns, regardless of the scaling of the \n",
      "    column.\n",
      "    \"\"\"\n",
      "    mi, ma = X.min(axis=0), X.max(axis=0)\n",
      "    col_neq = (ma - mi) > 1.e-8\n",
      "    Xs = np.ones_like(X)\n",
      "    mi = mi[col_neq]\n",
      "    ma = ma[col_neq]\n",
      "    Xs[:,col_neq] = (X[:,col_neq] - mi)/(ma - mi)\n",
      "    return Xs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def show_design(X, design_title):\n",
      "    \"\"\" Show the design matrix nicely \"\"\"\n",
      "    figure()\n",
      "    plt.gray()\n",
      "    imshow(scale_design_mtx(X), interpolation='nearest')\n",
      "    title(design_title)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# display using our fancy function\n",
      "show_design(X, 'Design matrix for the first analysis')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7O0JCQnDhwgXFVZEj4sjXjAoKCvD1118jOjq6zvOffPKJ9XGXLl3w61//ukXrslgssFgs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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x76790d0>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We need to know the degrees of freedom.  In this simple case, we have two *effects* (the intercept and the slope) and 12 observations, leaving us with 10 degrees of freedom.  In general, the number of *effects* we have is given by the number of independent columns in the design matrix $X$ above.  This is the *column rank* of $X$.  This is given by the numpy function ``numpy.linalg.matrix_rank`` for numpy >= 1.5, but in case you have an earlier version of numpy, a crude calculation of the matrix rank is a couple of lines of code:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def matrix_rank(X):\n",
      "    U, S, V = np.linalg.svd(X)\n",
      "    # Number of singular values above an arbitrary small threshold\n",
      "    return np.sum(S > 1e-5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we can calculate the degrees of freedom in a more general way:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "df = nimgs - matrix_rank(X)\n",
      "# mean SoS for guess slope\n",
      "guessRSS = (guessRes**2).sum() / df"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The analysis giving the least squares fit is very simple: we matrix multiply the pseudoinverse of $X$ (denoted $X^+$) by the data, to get the estimation of the model parameters, the $\\hat\\beta$ 's : $\\hat\\beta = X^+ Y$ "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# the analysis, giving slope and constant\n",
      "B = dot(pinv(X), Y)\n",
      "print B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[  0.63928135  54.39383796]\n"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# plot data with new least squares line\n",
      "plot(ourcov, Y, 'x')\n",
      "bound = np.array([minx, maxx])\n",
      "plot(bound, bound*B[0]+B[1], 'r')\n",
      "axis(axs)\n",
      "xlabel('Task difficulty')\n",
      "ylabel('PS voxel value')\n",
      "title('The least squares linear relationship');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BHV4URlWi1BO0wcHBOHLkCHr06AEA8PLyKrEl4e3bt+Hv7w8AyM3NxahRo9Cn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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x76916d0>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We'll need some statistical machinery from scipy"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "from scipy.stats import t as tdist, norm as normdist"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And now we can compute our stats for this contrast:\n",
      "The $\\textbf{Residual Sum of Square}$ (RSS) is :\n",
      "\n",
      "$$ RSS = \\sum_{i=1}^{12} (Y_i - X_i \\beta)^2$$\n",
      "\n",
      "$X_i$ is the $i$th row of the matrix $X$. The $\\textbf{Mean Residual Sum of Square}$ is :\n",
      "\n",
      "$$ MRSS = RSS / \\textit{df} $$\n",
      "\n",
      "where $\\textit{df}$ are the degrees of freedom, computed as the number of observations (here, 12) minus the rank of the design matrix (ie, the number of independent vectors in X). The $MRSS$ is the noise variance estimation: $\\widehat{\\sigma^2}$.\n",
      "\n",
      "The $\\textbf{Standard Error}$ (SE) of $c \\beta$ is computed with $\\textrm{SE} = \\sqrt{\\widehat{\\sigma^2} c (X^t X)^+ c^t} $ \n",
      "(why is that so is to be expanded here ...)\n",
      "\n",
      "The t test is then : $t = \\frac{c \\beta}{\\textrm{SE}} $"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Contrast\n",
      "C = np.array([1, 0])\n",
      "# t statistic and significance test\n",
      "RSS   = ((Y - dot(X, B))**2).sum()\n",
      "MRSS  = RSS / df\n",
      "SE    = np.sqrt(MRSS*dot(C, (dot(pinv(dot(X.T, X)), C.T))))\n",
      "t     = dot(C, B)/SE\n",
      "ltp   = tdist(df).cdf(t) # lower tail p\n",
      "p     = 1-ltp           # upper tail p \n",
      "\n",
      "# print results to window\n",
      "print 'First TD analysis: t= %2.2f, p= %0.6f' % (t, p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "First TD analysis: t= 7.95, p= 0.000006\n"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can package the analysis up into a function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def t_stat(Y, X, C):\n",
      "    \"\"\" betas, t statistic and significance test given data, design matrix, contrast\n",
      "    \"\"\"\n",
      "    # Calculate the parameters\n",
      "    B = dot(pinv(X), Y)\n",
      "    RSS   = ((Y - dot(X, B))**2).sum(axis=0)\n",
      "    # Recalculate df\n",
      "    df =  X.shape[0] - matrix_rank(X)\n",
      "    MRSS  = RSS / df\n",
      "    SE    = np.sqrt(MRSS*dot(C, (dot(pinv(dot(X.T, X)), C.T))))\n",
      "    t     = dot(C, B)/SE\n",
      "    ltp   = tdist(df).cdf(t) # lower tail p\n",
      "    p = 1 - ltp # upper tail p\n",
      "    return B, t, df, p\n",
      "\n",
      "# Results are the same\n",
      "print 'First TD analysis again: B = %s, t= %2.2f, df= %d, p= %0.6f' % t_stat(Y, X, C)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "First TD analysis again: B = [  0.63928135  54.39383796], t= 7.95, df= 10, p= 0.000006\n"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# save data for analysis in some package that reads csv type files\n",
      "np.savetxt(\"voxdata.txt\", Y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now the analysis for the added covariate.\n",
      "\n",
      "This and all the other analyses here use exactly the same code as above."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# design matrix, df\n",
      "X = np.column_stack([ourcov, np.arange(1, nimgs+1), np.ones(nimgs)])\n",
      "show_design(X, 'Design matrix for added covariate')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAN0AAAEICAYAAADbQPEyAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nUoyhI1KMoSNSjKEjUoyhI1KMoSNS7P8AC6HLOpLYAksAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x742d190>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Contrast\n",
      "C = np.array([1, 0, 0])\n",
      "# the analysis\n",
      "B, t, df, p = t_stat(Y, X, C)\n",
      "print 'Added covariate analysis: t= %2.2f, p= %0.6f' % (t, p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Added covariate analysis: t= 7.81, p= 0.000013\n"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Conditions analysis"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "X = np.zeros((nimgs, 2))\n",
      "X[0:6, 0] = 1\n",
      "X[6:, 1] = 1\n",
      "show_design(X, 'Design matrix for two conditions')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GRKQAQyJSgCERKcCQiBRgSEQKMCQiBRgSkQIMiUgBhkSkAEMiUoAhESnAkIgUYEhECjAk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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x76af950>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B = np.linalg.pinv(X).dot(Y)\n",
      "B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": [
        "array([ 56.54455457,  56.71809077])"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "These are the means for each set of observations (why?):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.mean(Y[:6]), np.mean(Y[6:])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "(56.544554566379446, 56.718090771077186)"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Contrast for activation minus rest\n",
      "C = np.array([-1, 1])\n",
      "# the analysis\n",
      "B, t, df, p = t_stat(Y, X, C)\n",
      "print 'Analysis for two conditions: t= %2.2f, p= %0.6f' % (t, p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Analysis for two conditions: t= 0.23, p= 0.409795\n"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Some extra analyses to show how it all works"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# do an ANCOVA for globals, instead of proportional scaling\n",
      "# GM scale\n",
      "Yanc = vdata * GM / np.mean(gdata)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = np.column_stack((ourcov, gdata, np.ones(nimgs)))\n",
      "show_design(X, 'Design with ANCOVA for global signal')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7187f10>"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Contrast for task covariate only\n",
      "C = np.array([1, 0, 0])\n",
      "# the analysis\n",
      "B, t, df, p = t_stat(Yanc, X, C)\n",
      "print 'TD analysis with ANCOVA for global: t= %2.2f, p= %0.6f' % (t, p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "TD analysis with ANCOVA for global: t= 7.15, p= 0.000027\n"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Another voxel"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# coordinates of voxel of interest in mm (MNI space)\n",
      "posmm2  = [6, 0, 6]\n",
      "# Coordinates in voxel space\n",
      "posvox2 = np.dot(iM, posmm2 + [1])\n",
      "# We grab the spatial part of the output.  Since we want to use it as an \n",
      "# index, we need to make it a tuple\n",
      "posvox2 = tuple(posvox2[:3].astype(int))\n",
      "posvox2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "(42, 56, 28)"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Get the data for this voxel\n",
      "vdata2 = np.zeros(nimgs)\n",
      "for i, V in enumerate(images):\n",
      "    d = V.get_data()\n",
      "    vdata2[i] = d[posvox2]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Proportional and global mean scale\n",
      "Y2 = vdata2 / gdata * 50"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = np.column_stack((ourcov, np.ones(nimgs)))\n",
      "C = np.array([1, 0])\n",
      "# the analysis\n",
      "B, t, df, p = t_stat(Y2, X, C)\n",
      "print 'First TD analysis second voxel: t= %2.2f, p= %0.6f' % (t, p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "First TD analysis second voxel: t= 5.84, p= 0.000082\n"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Analyzing two voxels at the same time, but independently:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Y_both = np.column_stack((Y, Y2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# the analysis\n",
      "B, t, df, p = t_stat(Y_both, X, C)\n",
      "print 'First TD analysis first voxel: t= %2.2f, p= %0.6f' % (t[0], p[0])\n",
      "print 'First TD analysis second voxel: t= %2.2f, p= %0.6f' % (t[1], p[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "First TD analysis first voxel: t= 7.95, p= 0.000006\n",
        "First TD analysis second voxel: t= 5.84, p= 0.000082\n"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "(2, 2)"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "You can see that the result when analyzing two voxels at the same time are identical to the results when analyzing them separately."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}